Calculus: Single Variable Part 3 – Integration Coursera Quiz Answers 2022 [💯Correct Answer]

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Calculus: Single Variable Part 3 - Integration Coursera Quiz Answers 2022 [💯Correct Answer]
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About Calculus: Single Variable Part 3 – Integration Course

In this third part, the third of five, we talk about how to integrate differential equations, how to integrate, the fundamental theorem of integral calculus, and how to integrate hard problems.

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Calculus: Single Variable Part 3 – Integration Quiz Answers

Week 01: Calculus: Single Variable Part 3 – Integration Coursera Quiz Answers

Main Quiz 01

Q1. \displaystyle \int (4x^3 + 3x^2 + 2x + 1) \, dx =∫(4x3+3x2+2x+1)dx=

  • 4x^4 + 3x^3 + 2x^2 + x + C4x4+3x3+2x2+x+C
  • \displaystyle \frac{1}{4}x^4 + \frac{1}{3}x^3 + \frac{1}{2}x^2 + x + C41​x4+31​x3+21​x2+x+C
  • x^4 + x^3 + x^2 + x + Cx4+x3+x2+x+C
  • 4x^3 + 3x^2 + x + C4x3+3x2+x+C
  • 12x^2 + 6x + 1 + C12x2+6x+1+C
  • 3x^2 + x + C3x2+x+C

Q2. \displaystyle \frac{d}{dx} \int \ln \tan x \, dx =dxd​∫lntanxdx=

  • \ln \tan x + Clntanx+C
  • \displaystyle \frac{\sec^2 x}{\tan x}tanxsec2x​
  • 00
  • \displaystyle \frac{\sec^2 x}{\tan x} + Ctanxsec2x​+C
  • \ln \tan xlntanx
  • \displaystyle \int \frac{\sec^2 x}{\tan x} \, dx∫tanxsec2x​dx

Q3. \displaystyle \int \left( \frac{d}{dx} e^{-x} \right) \, dx =∫(dxd​e−x)dx=

  • e^{x} + Cex+C
  • e^{-x}e−x
  • e^{-x} + Ce−x+C
  • \displaystyle \frac{d}{dx} \int e^{-x} \, dxdxd​∫e−xdx
  • -e^{-x} + C−e−x+C
  • -e^{-x}−e−x

Q4. Find the general solution of the differential equation

\frac{dx}{dt} = t^2dtdx​=t2

  • \displaystyle x(t) = \frac{1}{2} t^2 + Cx(t)=21​t2+C
  • x(t) = t^2 + Cx(t)=t2+C
  • \displaystyle x(t) = t^3 + Cx(t)=t3+C
  • \displaystyle x(t) = \frac{1}{3} t^3 + Cx(t)=31​t3+C
  • \displaystyle x(t) = \frac{1}{3} t^3x(t)=31​t3
  • \displaystyle x(t) = t^3x(t)=t3

Q5. Find the general solution of the differential equation

\frac{dx}{dt} = x^2dtdx​=x2

Hint: think in terms of differentials, as we did in the Lecture. This will allow you to bring everything that depends on xx to the left hand side, and everything that depends on tt to the right. Then integrate.

  • \displaystyle x(t) = -\frac{1}{t} + Cx(t)=−t1​+C
  • \displaystyle x(t) = \frac{1}{-t + C}x(t)=−t+C1​
  • \displaystyle x(t) = -\frac{1}{t^2} + Cx(t)=−t21​+C
  • \displaystyle x(t) = -\frac{1}{t}x(t)=−t1​
  • \displaystyle x(t) = -\frac{1}{t^2}x(t)=−t21​
  • \displaystyle x(t) = \frac{1}{-t^2 + C}x(t)=−t2+C1​

Practice Quiz 01

Q1. There is a large class of differential equations —the so-called linear ones— for which we can find solutions using the Taylor series method discussed in the Lecture. One such differential equation is

td2xdt2+dxdt+tx=0(∗)

It is a particular case of the more general Bessel differential equation, and one solution of it is given by the Bessel function J_0(t)J0​(t) that we saw in Chapter 1. Notice that (\ast∗) involves not only the first derivative \displaystyle \frac{dx}{dt}dtdx​ but also the second derivative \displaystyle \frac{d^2x}{dt^2}dt2d2x​. For this reason, it is said to be a second order differential equation.

In this problem we will content ourselves with finding a relationship (specifically, a recurrence relation) on the coefficients of a Taylor series expansion about t=0t=0 of a solution to our equation. Hence consider the Taylor series

x(t) = \sum_{k=0}^\infty c_k t^kx(t)=k=0∑∞​ck​tk

Substituting this into (\ast∗) will give you two conditions. The first one is c_1 = 0c1​=0. What is the other one?

Note: this problem involves some nontrivial manipulation of indices in summation notation. Do not get discouraged if it feels more difficult than other problems: it is!

  • \displaystyle c_k = \frac{c_{k-2}}{k^2}ck​=k2ck−2​​
  • \displaystyle c_k = \frac{c_{k-1}}{k^2}ck​=k2ck−1​​
  • \displaystyle c_k = – \frac{c_{k-1}}{(k-1)^2}ck​=−(k−1)2ck−1​​
  • \displaystyle c_k = – \frac{c_{k-2}}{k^2}ck​=−k2ck−2​​
  • \displaystyle c_k = – \frac{c_{k-2}}{(k-2)^2}ck​=−(k−2)2ck−2​​
  • \displaystyle c_k = – \frac{c_{k-1}}{k^2}ck​=−k2ck−1​​

Main Quiz 02

Q1. After drinking a cup of coffee, the amount CC of caffeine in a person’s body obeys the differential equation

\frac{dC}{dt}= -\alpha CdtdC​=−αC

where the constant \alphaα has an approximate value of 0.14 hours^{-1}−1.

How many hours will it take a human body to metabolize half of the initial amount of caffeine? Round your answer to the nearest integer.

Q2. The amount II of a radioactive substance in a given sample will decay in time according to the following equation: \frac{dI}{dt} = -\lambda IdtdI​=−λI Nuclear engineers and scientists tend to be concerned with the half-life of a substance, that is, the time it takes for the amount of radioactive material to be halved.

Find the half-life of a substance in terms of its decay constant \lambdaλ.

  • \displaystyle \frac{\ln 2}{\lambda}λln2​
  • \displaystyle \frac{2}{\lambda}λ2​
  • \displaystyle \frac{\lambda}{2}2λ​
  • \displaystyle \frac{\lambda}{\ln 2}ln2λ​
  • \displaystyle \frac{\ln \lambda}{2}2lnλ​
  • \displaystyle \frac{2}{\ln \lambda}lnλ2​

Q3. In a highly viscous fluid, a falling spherical object of radius rr decelerates right before reaching the bottom of the container. A simple model for this behavior is provided by the equation

\frac{dh}{dt} = – \frac{\alpha}{r}hdtdh​=−rα​h

where hh is the height of the object measured from the bottom, and \alphaα is a constant that depends on the viscosity of the fluid.

Find the time it would take the object to drop from h = 6rh=6r to h = 2rh=2r in terms of \alphaα and rr.

  • \displaystyle \frac{\alpha}{r} \ln \frac{1}{3}rα​ln31​
  • \displaystyle \frac{\alpha}{r} \ln 12rα​ln12
  • \displaystyle \frac{r}{\alpha} \ln 3αr​ln3
  • \displaystyle \frac{\alpha}{r} \ln 3rα​ln3
  • \displaystyle \frac{r}{\alpha} \ln \frac{1}{3}αr​ln31​
  • \displaystyle \frac{r}{\alpha} \ln 12αr​ln12

Q4. On a cold day you want to brew a nice hot cup of tea. You pour boiling water (at a temperature of 212^\mathrm{o} F212oF) into a mug and drop a tea bag in it. The water cools down in contact with the cold air according to Newton’s law of cooling:

\frac{dT}{dt} = \kappa (A – T)dtdT​=κ(A−T)

where TT is the temperature of the water, A = 32^\mathrm{o} FA=32oF the ambient temperature, and \kappa = 0.36κ=0.36 min^{-1}−1.

The threshold for human beings to feel pain when entering in contact with something hot is around 107^\mathrm{o} F107oF. How many seconds do you have to wait until you can safely take a sip? Round your answer to the nearest integer.

Q5. On the night of April 14, 1912, the British passenger liner RMS Titanic collided with an iceberg and sank in the North Atlantic Ocean. The ship lacked enough lifeboats to accommodate all of the passengers, and many of them died from hypothermia in the cold sea waters. Hypothermia is the condition in which the temperature of a human body drops below normal operating levels (around 36^\mathrm{o} C36oC). When the core body temperature drops below 28^\mathrm{o} C28oC, the hypothermia is said to have become severe: major organs shut down and eventually the heart stops.

If the water temperature that night was -2^\mathrm{o} C−2oC, how long did it take for passengers of the Titanic to enter severe hypothermia? Recall from lecture that heat transfer is described by Newton’s law of cooling:

\frac{dT}{dt} = \kappa (A – T)dtdT​=κ(A−T)

where TT is the body temperature of a passenger, AA the water temperature, and \kappa = 0.016κ=0.016 min^{-1}−1. Give your answer in minutes and round it to the nearest integer.

Practice Quiz 02

Q1. Assume the size PP of a population grows following the differential equation

\frac{dP(t)}{dt} = bP(t)dtdP(t)​=bP(t)

The rate of growth bb is the difference between the birth rate and the mortality rate. The Malthusian model supposes that this rate is constant. Of course we know that does not always happen: factors such as famines, outbreaks of disease or advances in medicine do influence these rates. When modeling some process mathematically it is important to recognize what our assumptions are and when they no longer hold. In this problem and the next we will look at one particular event that would result in a violation of Malthus’ assumption that the growth rate bb is constant over time: the occurrence of famines.

Experimental data suggests that food production FF grows linearly over time:

F(t) = F_0 + stF(t)=F0​+st

We will now make two assumptions:

  • Most food is perishable, so that the amount of food available at any given time is exactly the amount produced at that time. This means that we are not taking into account the effect of the possibility of preserving food for long periods of time.
  • The amount of food that a person in our population eats is, on average, constant and equal to some number \alphaα. That is, the amount of food necessary to keep everybody fed is \alpha P(t)αP(t).

The so-called Malthusian catastrophe happens when there is not enough food to feed the whole population, that is, when \alpha P(t) = F(t)αP(t)=F(t).

The Food and Agriculture Organization of the United Nations considers that a person needs around 1800 kcal/day to be considered well-fed. According to a report released in 2002 by the same organization:

  • The world population in 2002 was around 6 billion.
  • The population growth rate was estimated at 1.1% and expected to remain approximately constant for several decades.
  • Total food production in 2002 was determined to be around 6.13 \cdot 10^{15}6.13⋅1015 kcal, with an expected growth rate of 1.11 \cdot 10^{14}1.11⋅1014 kcal/year.

[Source: FAO, “World Agriculture: Towards 2015/2030. Summary Report”, 2002]

Estimate the world population by the year 2030.

  • 8.0 billion
  • 8.8 billion
  • 8.2 billion
  • 8.6 billion
  • 7.8 billion
  • 8.4 billion

Q2. [Continued from previous question] Estimate when the Malthusian catastrophe would happen if our assumptions continue to hold.

  • 39.42 e^{0.011 t} = 61.3 + 1.11 t \Rightarrow t \approx 16639.42e0.011t=61.3+1.11t⇒t≈166 years.
  • 39.42 e^{1.1 t} = 61.3 + 1.11 t \Rightarrow t \approx 0.439.42e1.1t=61.3+1.11t⇒t≈0.4 years.
  • 0.11 e^{1.1 t} = 61.3 + 1.11 t \Rightarrow t \approx 60.11e1.1t=61.3+1.11t⇒t≈6 years.
  • 0.11 e^{0.011 t} = 61.3 + 1.11 t \Rightarrow t \approx 8270.11e0.011t=61.3+1.11t⇒t≈827 years.

Main Quiz 03

Q1. Solve the differential equation \displaystyle \frac{dx}{dt} = \frac{x}{t}dtdx​=tx​.

Note: observe that this equation can be rearranged as \displaystyle \frac{dx}{x} = \frac{dt}{t}xdx​=tdt​, which says that the relatives rates of change of xx and tt are equal.

  • x(t) = Ctx(t)=Ct
  • x(t) = Ce^tx(t)=Cet
  • x(t) = \ln t + Cx(t)=lnt+C
  • x(t) = e^t + Cx(t)=et+C
  • x(t) = t + Cx(t)=t+C
  • x(t) = \ln(t + C)x(t)=ln(t+C

Q2. Solve the differential equation \displaystyle \frac{dx}{dt} = \frac{\sqrt{1-x^2}}{\sqrt{1-t^2}}dtdx​=1−t2​1−x2​​.

  • x(t) = \arcsin \sin(t + C)x(t)=arcsinsin(t+C)
  • x(t) = \sin ( \arcsin t + C )x(t)=sin(arcsint+C)
  • x(t) = \sin \arcsin(t + C)x(t)=sinarcsin(t+C)
  • x(t) = Ctx(t)=Ct
  • x(t) = t + Cx(t)=t+C
  • x(t) = \arcsin ( \sin t + C)x(t)=arcsin(sint+C)

Q3. Given that x(0)=0x(0)=0 and \displaystyle \frac{dx}{dt} = te^xdtdx​=tex, compute x(1)x(1).

  • x(1) = 0x(1)=0
  • x(1) = \sqrt{2}x(1)=2​
  • x(1) = \ln 2x(1)=ln2
  • x(1) = 2x(1)=2
  • x(1) = \displaystyle \ln \frac{1}{2}x(1)=ln21​
  • x(1) = \displaystyle\frac{1}{2}x(1)=21​

Q4. Which of the following is the integrating factor used to solve the following linear differential equation?

t^2\frac{dx}{dt}=4t-t^5xt2dtdx​=4t−t5x

  • I(t) = e^{-1/2t^2}I(t)=e−1/2t2
  • I(t) = e^{-t^6/6}I(t)=e−t6/6
  • I(t) = e^{t^6/6}I(t)=et6/6
  • I(t) = e^{1/2t^2}I(t)=e1/2t2
  • I(t) = e^{-t^4/4}I(t)=e−t4/4
  • I(t) = e^{t^4/4}I(t)=et4/4

Q5. Solve the differential equation \displaystyle \frac{dx}{dt} – 5x = 3dtdx​−5x=3.

  • \displaystyle x(t) = \frac{5}{3} + C e^{5t}x(t)=35​+Ce5t
  • \displaystyle x(t) = -\frac{3}{5} + C e^{-5t}x(t)=−53​+Ce−5t
  • \displaystyle x(t) = \frac{3}{5} + C e^{5t}x(t)=53​+Ce5t
  • \displaystyle x(t) = -\frac{3}{5} + C e^{5t}x(t)=−53​+Ce5t
  • \displaystyle x(t) = -\frac{5}{3} + C e^{-5t}x(t)=−35​+Ce−5t
  • \displaystyle x(t) = -\frac{5}{3} + C e^{5t}x(t)=−35​+Ce5t

Practice Quiz 03

Q1. Solve the differential equation \displaystyle \frac{dx}{dt} = \frac{x}{1+t} + 2dtdx​=1+tx​+2.

  • \displaystyle x(t) = 1 + t + \frac{C}{1+t}x(t)=1+t+1+tC​
  • x(t) = 2(1+t)\ln(1+t) + C(1+t)x(t)=2(1+t)ln(1+t)+C(1+t)
  • \displaystyle x(t) = C(1 + t) + \frac{1}{1+t}x(t)=C(1+t)+1+t1​
  • x(t) = 2C(1+t)\ln(1+t) + (1+t)x(t)=2C(1+t)ln(1+t)+(1+t)
  • x(t) = 2t(1+t) + C(1+t)x(t)=2t(1+t)+C(1+t)
  • \displaystyle x(t) = \frac{1 + t}{2} + \frac{C}{1+t}x(t)=21+t​+1+tC​

Q2. Suppose that, in order to buy a house, you obtain a mortgage. If the lender advertises an annual interest rate rr, your debt DD will increase exponentially according to the simple O.D.E.

\frac{dD}{dt} = rD.dtdD​=rD.

If you pay your debt at a rate of PP (continuous annual rate), the evolution of your debt will then (under assumptions of continual compounding and payment) obey the linear differential equation

\frac{dD}{dt} = rD – PdtdD​=rD−P

Using this model, answer the following question: if initial amount of the mortgage is for $400,000, the annual interest rate is 5%, and you pay at a rate of $40,000 every year, how many years will it take you to pay off the debt? Round your answer to the nearest integer

Q3. German physician Ernst Heinrich Weber (1795-1878) is considered one of the fathers of experimental psychology. In his study of perception, he noticed that the perceived difference between two almost-equal stimuli is proportional to the percentual difference between them. In terms of differentials, we can express Weber’s law as

dp = k \frac{dS}{S}dp=kSdS​

where pp is the perceived intensity of a stimulus and SS its actual strength. Observe that \displaystyle \frac{dS}{S}SdS​ is the relative rate of change of SS.

In what way must the magnitude of a stimulus change in time for a human being to perceive a linear growth?

  • Logarithmically.
  • Rationally.
  • Proportional to the square root.
  • Linearly.
  • Exponentially.
  • None of these.

Q4. Some nonlinear differential equations can be reduced to linear ones by a clever change of variables. Bernouilli equations

\frac{dx}{dt} + p(t) x = q(t) x^\alpha, \qquad \alpha \in \mathbb{R}dtdx​+p(t)x=q(t)xα,α∈R

constitute the most important case. Notice that for \alpha=0α=0 or \alpha=1α=1 the above equation is already linear. For other values of \alphaα, the substitution u = x^{1-\alpha}u=x1−α yields a linear differential equation in the variable uu.

Apply the above change of variables in the case

\frac{dx}{dt} + 2tx = x^3dtdx​+2tx=x3

to find the linear differential equations satisfied by uu.

  • \displaystyle \frac{du}{dt} – 4tu = -2dtdu​−4tu=−2
  • \displaystyle \frac{du}{dt} + 2tu = 1dtdu​+2tu=1
  • \displaystyle \frac{du}{dt} + 2tu = -2dtdu​+2tu=−2
  • \displaystyle \frac{du}{dt} – 4tu = 1dtdu​−4tu=1
  • \displaystyle \frac{du}{dt} = -2-2tdtdu​=−2−2t
  • \displaystyle \frac{du}{dt} = 1-2tdtdu​=1−2t

Main Quiz 04

Q1. For a differential equation

\frac{dx}{dt} = f(x)dtdx​=f(x)

with an equilibrium at x=0x=0, the linearization of the differential equation at the equilibrium is, as per our example in class,

\frac{dx}{dt} = f'(0)x ,dtdx​=f′(0)x,

and one hopes that the solution to the linearized equation provides a good approximation to the original, nonlinear equation. (It does, so long as f'(0)\neq 0f′(0)​=0).

The differential equation

\frac{dx}{dt} = (e^x-1)(x-1)dtdx​=(ex−1)(x−1)

has an equilibrium at x=0x=0. Which of the following is the linearized equation at this equilibrium?

  • \displaystyle \frac{dx}{dt} = -exdtdx​=−ex
  • \displaystyle \frac{dx}{dt} = 0dtdx​=0
  • \displaystyle \frac{dx}{dt} = e^xdtdx​=ex
  • \displaystyle \frac{dx}{dt} = xdtdx​=x
  • \displaystyle \frac{dx}{dt} = -xdtdx​=−x
  • \displaystyle \frac{dx}{dt} = exdtdx​=ex

Q2. The differential equation in Question 1 has another equilibrium at x=1x=1. Let’s look at that. Recall, for a differential equation

\frac{dx}{dt} = f(x)dtdx​=f(x)

with an equilibrium at x=ax=a, the linearization of the differential equation at the equilibrium is best expressed as a differential equation on a new variable, h=x-ah=x−a. You should think of hh as a small perturbation to the equilibrium. This perturbation changes at a rate given by the linearization at aa:

\frac{dh}{dt} = f'(a)h ,dtdh​=f′(a)h,

and one hopes that the solution to the linearized equation provides a good approximation for the original, nonlinear equation. (It does, so long as f'(a)\neq 0f′(a)​=0).

The differential equation

\frac{dx}{dt} = (e^x-1)(x-1)dtdx​=(ex−1)(x−1)

has an equilibrium at x=1x=1. What is the linearized equation at this equilibrium?

  • \displaystyle\frac{dh}{dt} = -hdtdh​=−h
  • \displaystyle\frac{dh}{dt} = (e-1)hdtdh​=(e−1)h
  • \displaystyle\frac{dh}{dt} = -ehdtdh​=−eh
  • \displaystyle\frac{dh}{dt} = ehdtdh​=eh
  • \displaystyle\frac{dh}{dt} = hdtdh​=h
  • \displaystyle\frac{dh}{dt} = 0dtdh​=0

Q3. It is easy to determine stability or instability of an equilibrium. Recall that an equilibrium x=ax=a for \displaystyle\frac{dx}{dt}=f(x)dtdx​=f(x) is stable if f'(a)\lt 0f′(a)<0 and is unstable if f'(a)\gt 0f′(a)>0.

Recall from Lecture 18, Newton’s Law of Heat Transfer, which states that

\frac{dT}{dt} = \kappa ( A – T ),dtdT​=κ(A−T),

where \kappa>0κ>0 is a thermal conductivity constant and AA is the (constant) ambient temperature. Find and classify the equilibria in this system.

  • T=-AT=−A is the unique equilibrium. It is unstable.
  • There are no equilibria, which is why the temperature never quite stops changing.
  • T=\kappaT=κ is the unique equilibrium. It is unstable.
  • T=\kappaT=κ is the unique equilibrium. It is stable.
  • T=AT=A is the unique equilibrium. It is stable.
  • There are equilibria at T=0T=0 (unstable) and T=AT=A (stable).

Q4. Consider the differential equation

\frac{dx}{dt} = f(x)dtdx​=f(x)

and assume that f(a)=0f(a)=0 for some constant aa.

Assume also that the function ff has a Taylor expansion at x = ax=a,

f(x) = b(x-a) + O\big((x-a)^2\big)f(x)=b(x−a)+O((x−a)2)

for some constant bb.

Which of the following statements about the differential equation are true? Select all that apply.

  • bb is a stable equilibrium if a \gt 0a>0.
  • bb is a stable equilibrium if a \lt 0a<0.
  • aa is a stable equilibrium if b \lt 0b<0.
  • bb is an unstable equilibrium if a \gt 0a>0.
  • aa is a stable equilibrium if b \gt 0b>0.
  • Neither aa nor bb is an equilibrium point.
  • aa is an unstable equilibrium if b \gt 0b>0.
  • aa is an unstable equilibrium if b \lt 0b<0.

Q5. Consider the differential equation

\frac{dy}{dt} = -2y + y^2 + y^3dtdy​=−2y+y2+y3

Find and classify all the equilibria.

  • y=0y=0: unstable. y=1y=1 and y=-2y=−2: unstable.
  • y=1y=1: stable. y=0y=0 and y=-2y=−2: unstable.
  • y=0y=0: stable. y=-1y=−1 and y=2y=2: unstable.
  • y=0y=0: stable. y=1y=1 and y=-2y=−2: unstable.
  • y=2y=2: unstable. y=0y=0 and y=-1y=−1: stable.
  • y=0y=0, y=1y=1, and y=-2y=−2: stable.

Q6. Recall from Lecture 19 how we computed the terminal velocity of a falling body with linear drag given by

m\frac{dv}{dt} = mg – \kappa v ,mdtdv​=mg−κv,

where, of course, mm is mass, gg is gravitation, vv is velocity, and \kappa>0κ>0 is the drag coefficient. Can you see how easily one can solve for the equilibrium v_\infty = mg/\kappav∞​=mg/κ?

Very good. Now, let’s use a more realistic model of drag that is quadratic as opposed to linear:

m\frac{dv}{dt} = mg – \lambda v^2 ,mdtdv​=mg−λv2,

where \lambda>0λ>0 is a constant drag coefficient. This differential equation is not as easy to solve (but soon you will learn how). Is there is terminal velocity? What is it?

  • \displaystyle v_\infty = \sqrt{\frac{mg}{\lambda}}v∞​=λmg​​ is an unstable equilibrium; there is no terminal velocity.
  • \displaystyle v_\infty = 0v∞​=0 is an unstable equilibrium; there is no terminal velocity.
  • \displaystyle v_\infty = 0v∞​=0 is an unstable equilibrium; this is the terminal velocity.
  • \displaystyle v_\infty = \frac{mg}{\lambda}v∞​=λmg​ is an unstable equilibrium; there is no terminal velocity.
  • \displaystyle v_\infty = \sqrt{\frac{mg}{\lambda}}v∞​=λmg​​ is a stable equilibrium; this is the terminal velocity.
  • \displaystyle v_\infty = \frac{mg}{\lambda}v∞​=λmg​ is a stable equilibrium; this is the terminal velocity.

Q7. Recall that with continuous compounding at an interest rate of r>0r>0, an investment I(t)I(t) with initial investment I_0=I(0)I0​=I(0) is I(t)=I_0e^{rt}I(t)=I0​ert. What happens if you wish to withdraw funds from the investment at a rate of spending SS, where S>0S>0 is constant? The differential equation is:

\frac{dI}{dt} = rI – SdtdI​=rI−S

Your goals are as follow. You have an initial investment I_0I0​, and you cannot change it or the rate rr. You want to be able to spend as much as possible but you also don’t want to ever spend all your money. What amount of spending rate SS can you bear?

Hint: If you’re not sure what to do, find and classify the equilibria in this model and think about which initial conditions lead to which long-term behaviors.

  • Make sure S \le I_0S≤I0​.
  • Make sure S \le rS≤r.
  • Any S \gt 0S>0 will lead to eventual loss of all funds.
  • Make sure S/r \le 1S/r≤1.
  • Make sure S \ge r I_0S≥rI0​
  • Make sure S \le r I_0S≤rI0​.

Practice Quiz 04

Q1. In our lesson, we looked at two oscillators with “sinusoidal” coupling. Other types of coupling are possible as well. Consider the system of two oscillators modeled by

\frac{d\theta_1}{dt} = 2 + \epsilon(e^{\theta_1-\theta_2}-1) \quad ; \quad \frac{d\theta_2}{dt} = 2 + \epsilon(1-e^{\theta_1-\theta_2})dtdθ1​​=2+ϵ(eθ1​−θ2​−1);dtdθ2​​=2+ϵ(1−eθ1​−θ2​)

Consider the phase difference \varphi = \theta_2 – \theta_1φ=θ2​−θ1​. Note that \varphi=0φ=0 (where the oscillators are coupled) is an equilibrium. What is the linearized equation for \varphiφ about 00?

This looks intimidating, but is very straightforward. If you’re not sure how to start, compute \displaystyle \frac{d\varphi}{dt}dtdφ​. Then linearize this about \varphi=0φ=0.

  • \displaystyle \frac{d\varphi}{dt}=\epsilon\varphidtdφ​=ϵφ
  • \displaystyle \frac{d\varphi}{dt}=-\epsilon\varphidtdφ​=−ϵφ
  • \displaystyle \frac{d\varphi}{dt}=2\epsilon\varphidtdφ​=2ϵφ
  • \displaystyle \frac{d\varphi}{dt}=\epsilondtdφ​=ϵ
  • \displaystyle \frac{d\varphi}{dt}=2\varphidtdφ​=2φ
  • \displaystyle \frac{d\varphi}{dt}=-2\epsilon\varphidtdφ​=−2ϵφ

Q2. Biology is full of oscillators. Consider the following biomechanical example. When you walk with your arms free, you may find that they swing back and forth in a periodic manner. Let’s model each arm as a simple oscillator obeying

\frac{d\theta_i}{dt} = adtdθi​​=a

where \theta_1θ1​ and \theta_2θ2​ encode the state of your arm. These are not the angle at which your arms reside; rather, these are abstract measurements of how far along each periodic swing your arm is. Your two arms are connected by your torso, which acts as a torsional spring. When you swing one arm out it imparts an impulse through your shoulders to the other arm (try walking around!).

Which of the following is a reasonable model for the coupling function FF between your two arms? Here, \epsilon>0ϵ>0 is a very small positive constant and the model is:

\frac{d\theta_1}{dt} = a + \epsilon F(\theta_2-\theta_1), \qquad \frac{d\theta_2}{dt} = a – \epsilon F(\theta_2-\theta_1)dtdθ1​​=a+ϵF(θ2​−θ1​),dtdθ2​​=a−ϵF(θ2​−θ1​)

Hint: I suggest you first derive the equation for phase difference \varphi=\theta_2-\theta_1φ=θ2​−θ1​ in terms of F(\varphi)F(φ). Then, consider what happens when you walk around. Do your arms become phase-locked somehow? What stable equilibrium for \varphiφ do you note?

  • F(\varphi)=-\sin(\varphi)F(φ)=−sin(φ)
  • F(\varphi)=\sin(\varphi)F(φ)=sin(φ)
  • F(\varphi)=\tan(\varphi)F(φ)=tan(φ)
  • F(\varphi)=-\cos(\varphi)F(φ)=−cos(φ)
  • F(\varphi)=\cos(\varphi)F(φ)=cos(φ)

Q3. For the brave: The following model of a biochemical “switch” comes from the (very pleasant) text of Strogatz, Nonlinear Dynamics and Chaos. Let G(t)G(t) denote a concentration of a certain gene product, where the gene is switched “on” or “off” depending on the presence of a chemical signal. The production of GG is influenced both by the chemical signal and by autocatalysis (a feedback mechanism) and decay (a natural degradation). After simplification, the behavior of GG follows:

\frac{dG}{dt} = a – bG – \frac{G}{1+G^2}dtdG​=a−bG−1+G2G​

where aa and bb are positive constants and G(t)\geq 0G(t)≥0.

The number and types of equilibria depend on the values of aa and bb. Unfortunately, solving for them analytically (that is, finding exact values) is… challenging. Without determining exactly where the equilibria are, what can you say about them?

  • For some values of aa and bb, there are no equilibria.
  • All equilibria are unstable.
  • There are at most three equilibria. When there are three, then two are stable and one is unstable.
  • There is always at least one equilibrium.
  • All equilibria are stable.

Week 02 :Calculus: Single Variable Part 3 – Integration Coursera Quiz Answers

Main Quiz 01

Q1. \displaystyle \int 3\cos x \, dx =∫3cosxdx=

  • -3\cos x + C−3cosx+C
  • \sin 3x + Csin3x+C
  • 3\sin x + C3sinx+C
  • \cos 3x + Ccos3x+C
  • -3\sin x + C−3sinx+C
  • 3\cos x + C3cosx+C

Q2. \displaystyle \int x \sec^2 x^2 \, dx =∫xsec2x2dx=

  • \displaystyle \frac{1}{2}x \sec^2 x^2 + C21​xsec2x2+C
  • \displaystyle \frac{1}{2} \tan x^2 + C21​tanx2+C
  • x^2 \tan x^2 + Cx2tanx2+C
  • 2 \sec x^2 + C2secx2+C
  • \displaystyle \frac{1}{2}x \tan^2 x^2 + C21​xtan2x2+C
  • \tan x^2 + Ctanx2+C

Q3. \displaystyle \int \frac{4x}{(x^2 – 1)^3} \, dx =∫(x2−1)34x​dx=

  • \displaystyle \frac{4}{3(x^2 – 1)^2} + C3(x2−1)24​+C
  • \displaystyle \frac{1}{(x^2 – 1)^2} + C(x2−1)21​+C
  • \displaystyle -\frac{1}{(x^2 – 1)^2} + C−(x2−1)21​+C
  • \displaystyle -\frac{2x}{(x^2 – 1)^2} + C−(x2−1)22x​+C
  • \displaystyle \frac{2}{3(x^2 – 1)^2} + C3(x2−1)22​+C
  • \displaystyle -\frac{2}{3(x^2 – 1)^2} + C−3(x2−1)22​+C

Q4. \displaystyle \int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx =∫x​ex​​dx=

  • e^{\sqrt{x} / 2} + Cex​/2+C
  • 2\sqrt{x} \, e^{\sqrt{x}} + C2x​ex​+C
  • \displaystyle \frac{1}{2}\sqrt{x} \, e^{\sqrt{x}} + C21​x​ex​+C
  • e^{2\sqrt{x}} + Ce2x​+C
  • \displaystyle e^{\sqrt{x}} + Cex​+C
  • 2e^{\sqrt{x}} + C2ex​+C

Q5. \displaystyle \int \frac{ \ln(15x^5)}{x} \, dx =∫xln(15x5)​dx=

  • \displaystyle \frac{5}{2} \ln(15x) + C25​ln(15x)+C
  • \displaystyle \frac{5}{2}x^6 +C25​x6+C
  • \displaystyle \frac{1}{60} \ln^2(15x^5) + C601​ln2(15x5)+C
  • 10\ln^2(15x) + C10ln2(15x)+C
  • \displaystyle \frac{1}{10} \ln^2(15x^5) + C101​ln2(15x5)+C
  • \displaystyle \frac{1}{10} \ln(15x) + C101​ln(15x)+C

Q6. \displaystyle \int \frac{x\, dx}{\sqrt{x+3}} =∫x+3​xdx​=

  • \displaystyle \frac{2}{3} (x+3)^{3/2} + C32​(x+3)3/2+C
  • \displaystyle \frac{1}{3} (x-3) \sqrt{x+3} + C31​(x−3)x+3​+C
  • \displaystyle \frac{1}{3} \left[ (x+3)^{3/2} – 2(x+3)^{1/2} \right] + C31​[(x+3)3/2−2(x+3)1/2]+C
  • \displaystyle \frac{2}{3}(x-6)\sqrt{x+3} + C32​(x−6)x+3​+C
  • \displaystyle \frac{1}{3} (x+6)\sqrt{x+3} + C31​(x+6)x+3​+C
  • \displaystyle \frac{2}{3} \left[ 2(x+3)^{3/2} + (x+3)^{1/2} \right] + C32​[2(x+3)3/2+(x+3)1/2]+C

Practice Quiz 01

Q1. Apply the substitution \displaystyle u = \frac{1}{x}u=x1​ to calculate the integral

I(x) = \int \frac{dx}{x\sqrt{x^2-1}}I(x)=∫xx2−1​dx​

assuming x > 0x>0.

  • \displaystyle I(x) = \arcsin \frac{1}{x} + CI(x)=arcsinx1​+C
  • \displaystyle I(x) = 2\sqrt{x^2-1} + CI(x)=2x2−1​+C
  • \displaystyle I(x) = -\arctan \frac{1}{x} + CI(x)=−arctanx1​+C
  • \displaystyle I(x) = \arctan \frac{1}{x} + CI(x)=arctanx1​+C
  • \displaystyle I(x) = -\arcsin \frac{1}{x} + CI(x)=−arcsinx1​+C
  • \displaystyle I(x) = -2\sqrt{x^2-1} + CI(x)=−2x2−1​+C

Q2. Apply the substitution u = \sqrt{x^2-1}u=x2−1​ to calculate the integral

I(x) = \int \frac{dx}{x\sqrt{x^2-1}}I(x)=∫xx2−1​dx​

Note: this is exactly the same integral as that of the previous problem! In Lecture 23, we will learn of yet another substitution that allows us to solve this integral.

  • \displaystyle I(x) = \arcsin\frac{1}{\sqrt{x^2-1}} + CI(x)=arcsinx2−1​1​+C
  • I(x) = 2\sqrt{x^2-1} + CI(x)=2x2−1​+C
  • I(x) = -\arctan\sqrt{x^2-1} + CI(x)=−arctanx2−1​+C
  • I(x) = \arctan\sqrt{x^2-1} + CI(x)=arctanx2−1​+C
  • I(x) = -2\sqrt{x^2-1} + CI(x)=−2x2−1​+C
  • \displaystyle I(x) = -\arcsin\frac{1}{\sqrt{x^2-1}} + CI(x)=−arcsinx2−1​1​+C

Main Quiz 02

Q1. \displaystyle \int x e^{x/2} \, dx =∫xex/2dx=

  • \displaystyle \frac{1}{4} x^2 e^{x/2} + C41​x2ex/2+C
  • 2e^{x/2} (x-1) + C2ex/2(x−1)+C
  • \displaystyle \frac{1}{2} e^{x/2} (x-2) + C21​ex/2(x−2)+C
  • \displaystyle \frac{1}{4} e^{x/2} (2x-1) + C41​ex/2(2x−1)+C
  • 2e^{x/2} (x-2) + C2ex/2(x−2)+C
  • e^{x^2/4} + Cex2/4+C

Q2. \displaystyle \int x^2e^{x/2} \, dx =∫x2ex/2dx=

  • e^{x/2} (x^2 – 2x + 4) + Cex/2(x2−2x+4)+C
  • e^{x/2} (x^2 – 4x – 8) + Cex/2(x2−4x−8)+C
  • 2e^{x/2} (x – 2) + C2ex/2(x−2)+C
  • 2e^{x/2} (x^2 – 4x – 8) + C2ex/2(x2−4x−8)+C
  • 2e^{x/2} (x^2 – 4x + 8) + C2ex/2(x2−4x+8)+C
  • \displaystyle e^{x/2} \left( \frac{1}{2}x^2 – \frac{1}{8}x + \frac{1}{16} \right) + Cex/2(21​x2−81​x+161​)+C

Q3. \displaystyle \int 3x \ln x \, dx =∫3xlnxdx=

  • \displaystyle \frac{3x^2}{2} \left( \ln x – 1 \right) + C23x2​(lnx−1)+C
  • \displaystyle \frac{x^2}{4} \left( 2\ln x – 1 \right) + C4x2​(2lnx−1)+C
  • \displaystyle \frac{3x^2}{4} \left( \ln x – 1 \right) + C43x2​(lnx−1)+C
  • \displaystyle \frac{x^2}{2} \left( 3 \ln x – x \right) + C2x2​(3lnx−x)+C
  • \displaystyle \frac{3x}{2} \left( 4 \ln x – 1 \right) + C23x​(4lnx−1)+C
  • \displaystyle \frac{3x^2}{4} \left( 2\ln x – 1 \right) + C43x2​(2lnx−1)+C

Q4. \displaystyle \int 3x^2 \ln x \, dx =∫3x2lnxdx=

  • \displaystyle x^3 \left( \ln x – \frac{1}{3} \right) + Cx3(lnx−31​)+C
  • \displaystyle \frac{1}{3}x^3( \ln x – 1) + C31​x3(lnx−1)+C
  • \displaystyle x ( \ln x – 1 ) + Cx(lnx−1)+C
  • \displaystyle x^2\left( \ln x – \frac{1}{2} \right) + Cx2(lnx−21​)+C
  • \displaystyle 3x^2\ln x + \frac{1}{4}x^4 + C3x2lnx+41​x4+C
  • \displaystyle 3x^2\ln x – \frac{1}{4}x^4 + C3x2lnx−41​x4+C

Q5. \displaystyle \int x^2 \cos\frac{x}{2} \, dx =∫x2cos2x​dx=

Hint: no mathematician ever remembers those reduction formulas in the Lecture, but we all remember that we get to them by repeated integration by parts.

  • \displaystyle (2x^2+16) \sin\frac{x}{2} – 8x\cos\frac{x}{2} + C(2x2+16)sin2x​−8xcos2x​+C
  • \displaystyle (16-2x^2) \sin\frac{x}{2} + 8x\cos\frac{x}{2} + C(16−2x2)sin2x​+8xcos2x​+C
  • \displaystyle (2x^2-16) \sin\frac{x}{2} – 8x\cos\frac{x}{2} + C(2x2−16)sin2x​−8xcos2x​+C
  • \displaystyle (2x^2+16) \sin\frac{x}{2} + 8x\cos\frac{x}{2} + C(2x2+16)sin2x​+8xcos2x​+C
  • \displaystyle (16-2x^2) \sin\frac{x}{2} – 8x\cos\frac{x}{2} + C(16−2x2)sin2x​−8xcos2x​+C
  • \displaystyle (2x^2-16) \sin\frac{x}{2} + 8x\cos\frac{x}{2} + C(2x2−16)sin2x​+8xcos2x​+C

Q6. \displaystyle \int e^{2x}\sin 3x \, dx =∫e2xsin3xdx=

  • \displaystyle \frac{2}{3}e^{2x}\sin 3x + \frac{1}{3}e^{2x}\cos 3x + C32​e2xsin3x+31​e2xcos3x+C
  • \displaystyle \frac{2}{3}e^{2x}\cos 3x + \frac{3}{13}e^{2x}\sin 3x + C32​e2xcos3x+133​e2xsin3x+C
  • \displaystyle \frac{2}{13}e^{2x}\sin 3x – \frac{3}{13}e^{2x}\cos 3x + C132​e2xsin3x−133​e2xcos3x+C
  • \displaystyle \frac{1}{2}e^{2x}\sin 3x – \frac{3}{4}e^{2x}\cos 3x + C21​e2xsin3x−43​e2xcos3x+C
  • \displaystyle \frac{2}{3}e^{2x}\sin 3x – \frac{4}{3}e^{2x}\cos 3x + C32​e2xsin3x−34​e2xcos3x+C
  • \displaystyle \frac{2}{13}e^{2x}\cos 3x – \frac{3}{13}e^{2x}\sin 3x + C132​e2xcos3x−133​e2xsin3x+C

Q7. Apply integration by parts with u = \ln xu=lnx and dv = dxdv=dx to find an antiderivative for the logarithm —that is, to solve the following integral:

I(x) = \int \ln x \, dxI(x)=∫lnxdx

  • I(x) = x \ln x + CI(x)=xlnx+C
  • I(x) = x \left( \ln x – 1 \right) + CI(x)=x(lnx−1)+C
  • \displaystyle I(x) = \frac{\ln^2 x}{2} + CI(x)=2ln2x​+C
  • \displaystyle I(x) = \frac{\ln^2 x}{x^2} + CI(x)=x2ln2x​+C
  • I(x) = x \left( \ln x – x \right) + CI(x)=x(lnx−x)+C
  • \displaystyle I(x) = \frac{1}{x} + CI(x)=x1​+C

Practice Quiz 02

Q1. \displaystyle \int \ln^2 x \, dx =∫ln2xdx=

  • \displaystyle \frac{1}{2}\ln^2 x – x\ln x + C21​ln2x−xlnx+C
  • \ln^2 x + x\ln x – 2x + Cln2x+xlnx−2x+C
  • \ln^2 x – 2x + Cln2x−2x+C
  • x \ln^2 x – \ln x + 2x + Cxln2x−lnx+2x+C
  • x \left( \ln^2 x – 2\ln x + 2 \right) + Cx(ln2x−2lnx+2)+C
  • x \left( \ln^2 x + \ln x – 2 \right) + Cx(ln2x+lnx−2)+C

Q2. \displaystyle \int \sin(\ln x) \, dx =∫sin(lnx)dx=

  • \displaystyle \frac{x}{2} \left[ \sin(\ln x) + \cos(\ln x)\right] + C2x​[sin(lnx)+cos(lnx)]+C
  • x \left[ \sin(\ln x) + \cos(\ln x)\right] + Cx[sin(lnx)+cos(lnx)]+C
  • 2x \sin(\ln x) + C2xsin(lnx)+C
  • x \left[ \sin(\ln x) – \cos(\ln x)\right] + Cx[sin(lnx)−cos(lnx)]+C
  • \displaystyle \frac{x}{2} \left[ \sin(\ln x) – \cos(\ln x)\right] + C2x​[sin(lnx)−cos(lnx)]+C
  • 2x \cos(\ln x) + C2xcos(lnx)+C

Q3. \displaystyle \int \arcsin 2x \, dx =∫arcsin2xdx=

  • \displaystyle x\arcsin 2x + \frac{1}{2} \sqrt{1 – x^2} + Cxarcsin2x+21​1−x2​+C
  • \displaystyle x\arcsin 2x + \frac{1}{2} \sqrt{1 – 4x^2} + Cxarcsin2x+21​1−4x2​+C
  • \displaystyle \arcsin 2x + \frac{1}{4} \sqrt{1 – 2x^2} + Carcsin2x+41​1−2x2​+C
  • 2x\arcsin 2x + 8 \sqrt{1 – x^2} + C2xarcsin2x+81−x2​+C
  • x\arcsin 2x + 2 \sqrt{1 – x^2} + Cxarcsin2x+21−x2​+C
  • \arcsin 2x + 2 \sqrt{1 – 4x^2} + Carcsin2x+21−4x2​+C

Main Quiz 03

Q1. Which of the following integrals do you get when performing the substitution x = \sin\thetax=sinθ in \displaystyle \int x^5 \sqrt{1-x^2} \, dx∫x51−x2​dx ?

Note: in Lecture 28 we will see how to attack integrals involving powers of trigonometric functions.

  • \displaystyle \int \sin^2\theta \cos^5 \theta \, d\theta∫sin2θcos5θdθ
  • \displaystyle – \int \sin^2\theta \cos^5 \theta \, d\theta−∫sin2θcos5θdθ
  • \displaystyle – \int \sin^5\theta \cos\theta \, d\theta−∫sin5θcosθdθ
  • \displaystyle \int \sin^5\theta \cos^2\theta \, d\theta∫sin5θcos2θdθ
  • \displaystyle – \int \sin^5\theta \cos^2\theta \, d\theta−∫sin5θcos2θdθ
  • \displaystyle \int \sin^5\theta \cos\theta \, d\theta∫sin5θcosθdθ

Q2. \displaystyle \int \frac{x^2}{\sqrt{4-x^2}} \, dx =∫4−x2​x2​dx=

  • \displaystyle -2 \arccos\frac{x}{2} – \frac{1}{2}x \sqrt{4-x^2} + C−2arccos2x​−21​x4−x2​+C
  • \displaystyle 2 \arcsin\frac{x}{2} – \frac{1}{2}x \sqrt{4-x^2} + C2arcsin2x​−21​x4−x2​+C
  • \displaystyle x – \frac{1}{4}x \sqrt{4 – x^2} + Cx−41​x4−x2​+C
  • \displaystyle -\frac{1}{2} \arccos\frac{x}{2} – \frac{1}{8}x \sqrt{4-x^2} + C−21​arccos2x​−81​x4−x2​+C
  • \displaystyle \frac{x}{2} – \frac{1}{4} \cos 2x + C2x​−41​cos2x+C
  • \displaystyle \frac{1}{2} \arcsin\frac{x}{2} – \frac{1}{8}x \sqrt{4-x^2} + C21​arcsin2x​−81​x4−x2​+C

Q3. In Lecture we saw that we can use the substitution \displaystyle x = \frac{b}{a}\sin\thetax=ab​sinθ to (hopefully) calculate integrals involving \sqrt{b^2 – a^2x^2}b2−a2x2​. Another equally suitable subsitution in this case is \displaystyle x = \frac{b}{a}\cos\thetax=ab​cosθ. Use the latter to compute

\displaystyle \int \frac {\sqrt{1-x^2}} {x^2} \, dx =∫x21−x2​​dx=

  • \displaystyle -\frac{ \sqrt{1-x^2}}{2x^2} – \frac{1}{2} \arccos x + C−2x21−x2​​−21​arccosx+C
  • \displaystyle -\frac{ \sqrt{1-x^2} }{x} + \arccos x + C−x1−x2​​+arccosx+C
  • \displaystyle -\frac{ \sqrt{1-x^2} }{x} – 2\arccos x + C−x1−x2​​−2arccosx+C
  • \displaystyle \frac{ \sqrt{1-x^2} }{x^2} + \arccos x + Cx21−x2​​+arccosx+C
  • x \sqrt{1-x^2} – \arccos x + Cx1−x2​−arccosx+C
  • \displaystyle \frac{ \sqrt{1-x^2} }{x^2} + 2\arccos x + Cx21−x2​​+2arccosx+C

Q4. \displaystyle \int (1-x^2)^{-3/2} \, dx =∫(1−x2)−3/2dx=

  • \displaystyle \arcsin x + \frac{x}{\sqrt{1-x^2}} + Carcsinx+1−x2​x​+C
  • \displaystyle \frac{x}{\sqrt{1-x^2}} + C1−x2​x​+C
  • \displaystyle -2\sqrt{1-x^2} + C−21−x2​+C
  • \displaystyle \frac{1}{2}\arcsin x + \sqrt{1-x^2} + C21​arcsinx+1−x2​+C
  • \displaystyle \arccos x + \frac{1}{\sqrt{1-x^2}} + Carccosx+1−x2​1​+C
  • \displaystyle \frac{1}{\sqrt{1-x^2}} + C1−x2​1​+C

Q5. \displaystyle \int \frac{x}{\sqrt{1+x^2}} \, dx =∫1+x2​x​dx=

  • \displaystyle \ln |x + 1| + \frac{1}{ \sqrt{1 + x^2}} + Cln∣x+1∣+1+x2​1​+C
  • \displaystyle \frac{x}{1 + x^2} + C1+x2x​+C
  • \sqrt{1+x^2} + C1+x2​+C
  • \displaystyle \frac{1}{\sqrt{1+x^2}} + C1+x2​1​+C
  • \ln \big| x + \sqrt{1+x^2} \big| + Cln∣∣∣​x+1+x2​∣∣∣​+C
  • \displaystyle \frac{1}{2} \left( \mathrm{arcsinh}\, x – x \sqrt{1+x^2}\right) + C21​(arcsinhx−x1+x2​)+C

Q6. For the appropriate value of aa, the substitution x = a \sec\thetax=asecθ helps in reducing the integral \displaystyle \int \frac{x^4 \, dx}{\sqrt{x^2 – 4}}∫x2−4​x4dx​ to an integral involving trigonometric functions —which we will see how to deal with in Lecture 28. What value should you take for aa, and what integral do you get after performing the substitution?

  • For a = 2a=2, we obtain the integral \displaystyle \int 2^{10} \sec^5\theta \, d\theta∫210sec5θdθ
  • For a = 2a=2, we obtain the integral \displaystyle \int 2^4 \sec^5\theta \, d\theta∫24sec5θdθ
  • For a = 2a=2, we obtain the integral \displaystyle \int 2^5 \sec^5\theta \, d\theta∫25sec5θdθ
  • For a = 4a=4, we obtain the integral \displaystyle \int 2^8 \sec^4\theta \, d\theta∫28sec4θdθ
  • For a = 2a=2, we obtain the integral \displaystyle \int 2^4 \sec^4\theta \tan\theta \, d\theta∫24sec4θtanθdθ
  • For a = 4a=4, we obtain the integral \displaystyle \int 2^8 \sec^5\theta \, d\theta∫28sec5θdθ

Practice Quiz 03

Q1. In Homework 21 (challenge), we use the substitutions \displaystyle u = \frac{1}{x}u=x1​ and u = \sqrt{x^2-1}u=x2−1​ to compute the integral

I(x) = \int \frac{dx}{x\sqrt{x^2-1}}I(x)=∫xx2−1​dx​

Another possible substitution is x = \sec\thetax=secθ. Using it, compute I(x)I(x).

  • \displaystyle \frac{\sqrt{x^2-1}}{2x^2} + C2x2x2−1​​+C
  • \displaystyle \frac{1}{2}\sqrt{x^2-1} – \mathrm{arcsec}\, x + C21​x2−1​−arcsecx+C
  • x \arccos \sqrt{x^2-1} + Cxarccosx2−1​+C
  • \displaystyle \sqrt{x^2-1}\arccos x + Cx2−1​arccosx+C
  • \displaystyle \mathrm{arcsec}\, \frac{1}{x} + Carcsecx1​+C
  • \displaystyle \arccos \frac{1}{x} + Carccosx1​+C

Q2. \displaystyle \int \frac{dx}{\sqrt{x^2-6x+10}} =∫x2−6x+10​dx​=

(there may be multiple correct answers)

  • \mathrm{arcsinh}\, (x-3) + Carcsinh(x−3)+C
  • \displaystyle \frac{1}{2} \mathrm{arcsinh}\,(x-3) + \frac{1}{4} x \sqrt{x^2-6x+10} + C21​arcsinh(x−3)+41​xx2−6x+10​+C
  • \displaystyle \frac{\sqrt{x^2-6x+10}}{2x} + C2xx2−6x+10​​+C
  • \mathrm{arcsinh}\,\sqrt{x^2-6x+10} + Carcsinhx2−6x+10​+C
  • \ln \big| x – 3 + \sqrt{x^2 – 6x + 10}\big| + Cln∣∣∣​x−3+x2−6x+10​∣∣∣​+C
  • \displaystyle \frac{1}{2}\mathrm{arccosh}\, (x-3) – \frac{\sqrt{x^2-6x+10}}{4x} + C21​arccosh(x−3)−4xx2−6x+10​​+C

Q3. \displaystyle \int \frac{dx}{\sqrt{x^2-2x-8}} =∫x2−2x−8​dx​=

(there may be multiple correct answers)

  • \displaystyle \frac{1}{2}\sqrt{x^2-2x-8} + \mathrm{arccosh}\frac{x-1}{3} + C21​x2−2x−8​+arccosh3x−1​+C
  • \displaystyle -\sqrt{x^2-2x-8} + C−x2−2x−8​+C
  • \displaystyle \ln \big| x-1 + \sqrt{x^2-2x-8} \big| + Cln∣∣∣​x−1+x2−2x−8​∣∣∣​+C
  • \displaystyle \mathrm{arccosh}\frac{x-1}{3} + Carccosh3x−1​+C
  • \displaystyle \frac{1}{3}\ln \left| x – 1 + \sqrt{x^2-2x-8} \right| + C31​ln∣∣∣∣​x−1+x2−2x−8​∣∣∣∣​+C
  • \displaystyle \frac{1}{3}\sqrt{x^2-2x-8} – \mathrm{arccosh}\frac{x-1}{3} + C31​x2−2x−8​−arccosh3x−1​+C

Main Quiz 04

Q1. \displaystyle \int \frac{5+x}{x^2+x-6} \, dx =∫x2+x−65+x​dx=

  • \displaystyle \frac{1}{5} \ln \left| \frac{x +3}{x – 2} \right| + C51​ln∣∣∣∣∣​x−2x+3​∣∣∣∣∣​+C
  • \ln |x – 2| – \ln |x+3| + Cln∣x−2∣−ln∣x+3∣+C
  • \displaystyle \frac{7}{5} \ln \left| \frac{x – 2}{x+3} \right| + C57​ln∣∣∣∣∣​x+3x−2​∣∣∣∣∣​+C
  • \displaystyle \ln |x^2 – x + 6| + \frac{11}{\sqrt{23}} \arctan\frac{2x – 1}{\sqrt{23}} + Cln∣x2−x+6∣+23​11​arctan23​2x−1​+C
  • \displaystyle \frac{2}{5} \ln |x – 2| – \frac{7}{5} \ln |x+3| + C52​ln∣x−2∣−57​ln∣x+3∣+C
  • \displaystyle \frac{7}{5} \ln |x – 2| – \frac{2}{5} \ln |x+3| + C57​ln∣x−2∣−52​ln∣x+3∣+C

Q2. \displaystyle \int \frac{2x + 3}{6x^2 + 5x + 1} \, dx =∫6x2+5x+12x+3​dx=

  • \displaystyle \frac{7}{3}\ln |3x + 1| – \frac{2}{3}\ln |2x + 1| + C37​ln∣3x+1∣−32​ln∣2x+1∣+C
  • \displaystyle \frac{7}{3}\ln |2x + 1| – 4\ln |3x + 1| + C37​ln∣2x+1∣−4ln∣3x+1∣+C
  • \displaystyle 7\ln |3x + 1| – \frac{2}{3}\ln |2x + 1| + C7ln∣3x+1∣−32​ln∣2x+1∣+C
  • \displaystyle \frac{7}{3}\ln |3x + 1| – 2\ln |2x + 1| + C37​ln∣3x+1∣−2ln∣2x+1∣+C
  • \displaystyle \frac{7}{3}\ln |2x + 1| – 2\ln |3x + 1| + C37​ln∣2x+1∣−2ln∣3x+1∣+C
  • 7\ln |3x + 1| – 2\ln |2x + 1| + C7ln∣3x+1∣−2ln∣2x+1∣+C

Q3. \displaystyle \int \frac{x^2-x+5}{(x-2)(x-1)(x+3)} \, dx =∫(x−2)(x−1)(x+3)x2−x+5​dx=

  • \displaystyle \frac{2}{5} \ln |x – 2| – \frac{3}{4} \ln |x-1| + C52​ln∣x−2∣−43​ln∣x−1∣+C
  • \displaystyle -\frac{1}{4} \ln \left| \frac{x^2 + x – 6}{x-1} \right| + C−41​ln∣∣∣∣∣​x−1x2+x−6​∣∣∣∣∣​+C
  • \displaystyle \frac{7}{5} \ln |x – 2| – \ln |x – 1| + \frac{1}{20} \ln |x + 3| + C57​ln∣x−2∣−ln∣x−1∣+201​ln∣x+3∣+C
  • \displaystyle \frac{7}{5} \ln |x – 2| – \frac{5}{4} \ln |x – 1| + \frac{17}{20} \ln |x + 3| + C57​ln∣x−2∣−45​ln∣x−1∣+2017​ln∣x+3∣+C
  • \displaystyle \frac{1}{4} \ln \left| \frac{x^2 + x – 6}{x-1} \right| + C41​ln∣∣∣∣∣​x−1x2+x−6​∣∣∣∣∣​+C
  • \displaystyle \frac{37}{43} \ln |(x – 2)(x – 1)(x + 3)| + C4337​ln∣(x−2)(x−1)(x+3)∣+C

Q4. \displaystyle \int \frac{2x-1}{x^3-x} \, dx =∫x3−x2x−1​dx=

  • \displaystyle \ln \left| \frac{x(x-1)}{x+1} \right| + Cln∣∣∣∣∣​x+1x(x−1)​∣∣∣∣∣​+C
  • \displaystyle \ln |x| – \frac{1}{2} \ln |x-2| – \frac{1}{2} \ln |x+1| + Cln∣x∣−21​ln∣x−2∣−21​ln∣x+1∣+C
  • \displaystyle \ln \left| \frac{x^2(x-1)}{(x+1)^3} \right| + Cln∣∣∣∣∣​(x+1)3x2(x−1)​∣∣∣∣∣​+C
  • \displaystyle \ln \left| \frac{x+1}{x(x-1)} \right| + Cln∣∣∣∣∣​x(x−1)x+1​∣∣∣∣∣​+C
  • \displaystyle \ln |x| + \frac{1}{2}\ln|x-1| – \frac{3}{2}\ln |x+1| + Cln∣x∣+21​ln∣x−1∣−23​ln∣x+1∣+C
  • \displaystyle \ln \left| \frac{(x+1)^3}{x^2(x-1)} \right| + Cln∣∣∣∣∣​x2(x−1)(x+1)3​∣∣∣∣∣​+C

Q5. \displaystyle{\int} \frac{x^2-3}{x^2-4} \, dx =∫x2−4x2−3​dx=

Hint: start by performing long division of the numerator by the denominator.

  • \displaystyle x+\frac{1}{4} \ln|x^2-4|+Cx+41​ln∣x2−4∣+C
  • \displaystyle x + \frac{1}{4}\ln\left|\frac{x-2}{x+2}\right| + Cx+41​ln∣∣∣∣∣​x+2x−2​∣∣∣∣∣​+C
  • \displaystyle \frac{3}{4} \ln\left|\frac{x+2}{x-2}\right|+C43​ln∣∣∣∣∣​x−2x+2​∣∣∣∣∣​+C
  • \displaystyle x-\frac{1}{x}-4x+Cx−x1​−4x+C
  • \displaystyle x-\frac{3}{4} \ln\left|\frac{x-2}{x+2}\right|+Cx−43​ln∣∣∣∣∣​x+2x−2​∣∣∣∣∣​+C
  • \displaystyle x + \frac{1}{2}\ln|x-2| – \frac{1}{4}\ln|x+2| + Cx+21​ln∣x−2∣−41​ln∣x+2∣+C

Q6. \displaystyle \int \frac{dx}{x^2 – 4x + 8} =∫x2−4x+8dx​=

Hint: complete the square in the denominator and perform a judicious substitution to get an integral of the form

\displaystyle \int \frac{du}{1+u^2}∫1+u2du​.

  • \displaystyle \frac{1}{4} \arctan \frac{x+2}{3} + C41​arctan3x+2​+C
  • \displaystyle \frac{1}{2} \arctan \frac{x-2}{2} + C21​arctan2x−2​+C
  • \displaystyle \frac{1}{2} \arctan \left(1 + \frac{x}{2}\right) + C21​arctan(1+2x​)+C
  • \displaystyle \frac{1}{4} \arctan \left(1 + \frac{x}{2}\right) + C41​arctan(1+2x​)+C
  • \displaystyle \frac{1}{4} \arctan \frac{x-2}{2} + C41​arctan2x−2​+C
  • \displaystyle \frac{1}{2} \arctan \frac{x+2}{3} + C21​arctan3x+2​+C

Practice Quiz 04

Q1. \displaystyle \int \frac{x^3+10x^2+33x+36}{x^2+4x+3} \, dx =∫x2+4x+3x3+10x2+33x+36​dx=

  • \ln |x + 1| – \ln |x+3| + Cln∣x+1∣−ln∣x+3∣+C
  • \displaystyle \frac{8}{9} \ln |x + 1| + \frac{4}{9} \ln |x+3| + C98​ln∣x+1∣+94​ln∣x+3∣+C
  • \displaystyle \frac{1}{2}x^2 + 6x + 6 \ln |x+1| + C21​x2+6x+6ln∣x+1∣+C
  • \displaystyle \frac{1}{2}x^2 + \ln |x^2 + 4x + 3| + C21​x2+ln∣x2+4x+3∣+C
  • x^2 + 6 \ln |x+1| + Cx2+6ln∣x+1∣+C
  • \displaystyle \frac{1}{2}x^2 + 6x + 3 \ln |x+1| + 3 \ln|x+3|+ C21​x2+6x+3ln∣x+1∣+3ln∣x+3∣+C

Q2. \displaystyle \int \frac{x+2}{(x-1)^2} \, dx =∫(x−1)2x+2​dx=

Hint: remember from the Lecture that we can deal with multiple roots in the denominator by using a partial fraction decomposition of the form

\frac{P(x)}{(x-r)^n} = \frac{A_1}{x-r} + \frac{A_2}{(x-r)^2} + \cdots + \frac{A_n}{(x-r)^n}(x−r)nP(x)​=x−rA1​​+(x−r)2A2​​+⋯+(x−r)nAn​​

Clearing denominators results in the equation:

P(x) = A_1 (x-r)^{n-1} + A_2 (x-r)^{n-2} + \cdots + A_n \tag{$\ast$}P(x)=A1​(x−r)n−1+A2​(x−r)n−2+⋯+An​(∗)

Notice that, in this case, the direct technique for finding the coefficients A_1, \ldots, A_nA1​,…,An​ —substituting x=rx=r in (\ast∗)— does not quite work: it only gives you the last coefficient, A_nAn​. In order to find the other coefficients, A_1, \ldots, A_{n-1}A1​,…,An−1​, you can revert to equating the coefficients of each power of xx on both sides of the equation (\ast∗).

  • \displaystyle \ln |x-1| + \frac{3}{x-1} + Cln∣x−1∣+x−13​+C
  • \displaystyle \ln |x-1| – \frac{3}{x-1} + Cln∣x−1∣−x−13​+C
  • 3\ln|x+1| + C3ln∣x+1∣+C
  • \displaystyle \ln |x-1| + \frac{2}{x-1} + Cln∣x−1∣+x−12​+C
  • \ln|x+1| + 3\ln|x+1|^2 + Cln∣x+1∣+3ln∣x+1∣2+C
  • \ln|x+1| – 3\ln|x+1|^2 + Cln∣x+1∣−3ln∣x+1∣2+C

Q3. Consider the integral

I(x) = \int \frac{dx}{x^4 – 6x^3 + 12x^2}I(x)=∫x4−6x3+12x2dx​

Factoring the denominator will get you a repeated linear factor —which we have just seen how to deal with— and an irreducible quadratic polynomial —which we know leads to an arctangent (and maybe something more). Now just put everything together to solve this integral!

Hint: when trying to solve for the coefficients of the partial fraction decomposition (there should be four of them), you might feel a little bit overwhelmed. Don’t worry! Start by plugging in x=0x=0 —that should give you one of the coefficients. To find the rest of them, remember that two polynomials are equal if and only if the coefficients of each of the terms are equal. That is,

1 = a x^3 + b x^2 + c x + d1=ax3+bx2+cx+d

if and only if a=0a=0, b=0b=0, c=0c=0 and d=1d=1.

  • \displaystyle I(x) = \frac{1}{36x} \left[ \sqrt{3} x \arctan\frac{x-3}{\sqrt{3}} +3\right] + CI(x)=36x1​[3​xarctan3​x−3​+3]+C
  • \displaystyle I(x) = -\frac{1}{36x} \left[ \sqrt{3} x \arctan\frac{x-3}{\sqrt{3}} +3\right] + CI(x)=−36x1​[3​xarctan3​x−3​+3]+C
  • \displaystyle I(x) = \frac{1}{24} \left[ \ln |x| – \frac{2}{x} – \frac{1}{2}\ln\left(x^2-6x+12\right) – \frac{1}{\sqrt{3}} \arctan\frac{x-3}{\sqrt{3}} \right] + CI(x)=241​[ln∣x∣−x2​−21​ln(x2−6x+12)−3​1​arctan3​x−3​]+C
  • \displaystyle I(x) = \frac{1}{24} \left[ \ln |x| – \frac{2}{x} – \frac{1}{2}\ln\left(x^2-6x+12\right) + \frac{1}{\sqrt{3}} \arctan\frac{x-3}{\sqrt{3}} \right] + CI(x)=241​[ln∣x∣−x2​−21​ln(x2−6x+12)+3​1​arctan3​x−3​]+C
  • \displaystyle I(x) = \frac{1}{48} \left[2 \ln |x| – \frac{1}{x} – \frac{1}{2}\ln\left(x^2-6x+12\right) – \frac{2}{3} \arctan{(x-3)} \right] + CI(x)=481​[2ln∣x∣−x1​−21​ln(x2−6x+12)−32​arctan(x−3)]+C
  • \displaystyle I(x) = \frac{1}{24} \left[ \ln |x| – \frac{2}{x} + \frac{1}{2}\ln\left(x^2-6x+12\right) -\frac{1}{\sqrt{3}} \arctan\frac{x-3}{\sqrt{3}} \right] + CI(x)=241​[ln∣x∣−x2​+21​ln(x2−6x+12)−3​1​arctan3​x−3​]+C

Week 03:Calculus: Single Variable Part 3 – Integration Coursera Quiz Answers

Main Quiz 01

Q1. Recall the definition of definite integrals through Riemann sums:

\int_{x=a}^b f(x)\, dx = \lim_{P: \Delta x \to 0} \sum_{i = 1}^n f(x_i) (\Delta x)_i∫x=ab​f(x)dx=P:Δx→0lim​i=1∑n​f(xi​)(Δx)i​

Here PP is a partition of the interval [a, b][a,b] into nn intervals P_iPi​, each of width (\Delta x)_i(Δx)i​. The point x_ixi​ is a sampling of P_iPi​, that is, a point in the interval P_iPi​. The limit is taken over all partitions as the width of the subdivisions gets smaller and smaller.

One particular choice of partition and sampling that can be used to numerically evaluate definite integrals is the following. With nn fixed, divide the interval [a, b][a,b] into nn subintervals P_iPi​ of common length (\Delta x)_i = (b-a)/n(Δx)i​=(b−a)/n. For the sampling, choose the right endpoint of each P_iPi​; this gives you the formula

x_i = a + i \frac{b-a}{n}xi​=a+inb−a​

With these choices of partition and sampling, compute the Riemann sums for the integral

\int_{x=1}^2 \frac{dx}{x}∫x=12​xdx​

for n=1n=1, n=2n=2 and n=3n=3 subdivisions.

Note: in the next Lecture we will learn that

\int_{x=1}^2 \frac{dx}{x} = \ln 2 \simeq 0.693∫x=12​xdx​=ln2≃0.693

Compare this value to the ones you obtain from the Riemann sums.

  • \displaystyle 1, \frac{3}{2} = 1.5, \frac{11}{6} \simeq 1.8331,23​=1.5,611​≃1.833
  • \displaystyle \frac{1}{2} = 0.5, \frac{5}{6} \simeq 0.833, \frac{13}{12} \simeq 1.08321​=0.5,65​≃0.833,1213​≃1.083
  • \displaystyle 1, \frac{1}{2} = 0.5, \frac{5}{6} \simeq 0.8331,21​=0.5,65​≃0.833
  • \displaystyle \frac{1}{2} = 0.5, \frac{7}{12} \simeq 0.583, \frac{37}{60} \simeq 0.61721​=0.5,127​≃0.583,6037​≃0.617
  • \displaystyle \frac{1}{2} = 0.5, \frac{5}{6} \simeq 0.833, \frac{5}{12} \simeq 0.41721​=0.5,65​≃0.833,125​≃0.417
  • \displaystyle 1, \frac{5}{6} \simeq 0.833, \frac{47}{60} \simeq 0.7831,65​≃0.833,6047​≃0.783

Q2. With the same choices of partition and sampling as in the previous problem, evaluate the Riemann sum for the integral

\int_{x=0}^3 x^2 \, dx∫x=03​x2dx

for an arbitrary number nn of subdivisions.

You might need to use any of the following formulas:

\sum_{i=1}^n i = \frac{n(n+1)}{2}, \quad \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}, \quad \sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}i=1∑n​i=2n(n+1)​,i=1∑n​i2=6n(n+1)(2n+1)​,i=1∑n​i3=4n2(n+1)2​

  • \displaystyle 9 + \frac{9}{n} + \frac{9}{n^2}9+n9​+n29​
  • \displaystyle 9 + \frac{9}{2n} + \frac{3}{2n^2}9+2n9​+2n23​
  • \displaystyle 9 + \frac{2n}{9} + \frac{2n^2}{3}9+92n​+32n2​
  • \displaystyle 9 + \frac{27}{2n} + \frac{9}{2n^2}9+2n27​+2n29​
  • \displaystyle 9 + \frac{2n}{27} + \frac{2n^2}{9}9+272n​+92n2​
  • 99

Q3. The line y=xy=x, the xx-axis and the vertical line x=2x=2 bound a triangle of area 22. Thus,

I = \int_{x=0}^2 x \, dx = 2I=∫x=02​xdx=2

Evaluating the Riemann sum for nn subdivisions for the above integral with the same choices of partition and sampling as in the previous problem yields an approximation RS(n)RS(n) for its value II. The error E(n)E(n) we commit by using this approximation is defined to be the difference

E(n) = RS(n) – IE(n)=RS(n)−I

Some of the following statements are true, while others are false. Among the correct ones, which one is optimal (assuming that you want to know how bad the error can be)?

  • E(n)E(n) is in O(1/n^3)O(1/n3) as n \to +\inftyn→+∞
  • E(n)E(n) is in O(n^2)O(n2) as n \to +\inftyn→+∞
  • E(n)E(n) is in O(1/n)O(1/n) as n \to +\inftyn→+∞
  • E(n)E(n) is in O(1/n^2)O(1/n2) as n \to +\inftyn→+∞
  • E(n)E(n) is in O(n)O(n) as n \to +\inftyn→+∞
  • E(n)E(n) is in O(1)O(1) as n \to +\inftyn→+∞

Q4. Evaluate \displaystyle \int_{x=-\pi/4}^{\pi/4} \left( x^2 + \ln|\cos x|\right) \sin \frac{x}{2} \, dx∫x=−π/4π/4​(x2+ln∣cosx∣)sin2x​dx. Provide a numeric answer.

Hint: before you try to compute anything, think

Q5. Every function can be expressed as the sum of an even function —called its even part— and an odd function —its odd part. They are given by the formulas

f^{\mathrm{even}}(x) = \frac{f(x) + f(-x)}{2}, \qquad f^{\mathrm{odd}}(x) = \frac{f(x) – f(-x)}{2}feven(x)=2f(x)+f(−x)​,fodd(x)=2f(x)−f(−x)​

Notice that, indeed,

f^{\mathrm{even}}(-x) = \frac{f(-x) + f(x)}{2} = f^{\mathrm{even}}(x), \qquad f^{\mathrm{odd}}(-x) = \frac{f(-x) – f(x)}{2} = -f^{\mathrm{odd}}(x)feven(−x)=2f(−x)+f(x)​=feven(x),fodd(−x)=2f(−x)−f(x)​=−fodd(x)

and

f^{\mathrm{even}}(x) + f^{\mathrm{odd}}(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) – f(-x)}{2} = f(x)feven(x)+fodd(x)=2f(x)+f(−x)​+2f(x)−f(−x)​=f(x)

If f(x) = e^xf(x)=ex, which of the following statements are true? Select all that apply.

  • \displaystyle f^{\mathrm{odd}}(x) = 0fodd(x)=0
  • \displaystyle f^{\mathrm{even}}(x) = \sinh xfeven(x)=sinhx
  • \displaystyle f^{\mathrm{odd}}(x) = \sinh xfodd(x)=sinhx
  • \displaystyle f^{\mathrm{even}}(x) = 0feven(x)=0
  • \displaystyle f^{\mathrm{even}}(x) = \cosh xfeven(x)=coshx
  • \displaystyle f^{\mathrm{odd}}(x) = \cosh xfodd(x)=coshx

Practice Quiz 01

Q1. \displaystyle \frac{d}{dx} \int_{t=0}^{\arcsin x} \ln \left| \sin t + \cos t \right| \, dt =dxd​∫t=0arcsinx​ln∣sint+cost∣dt=

  • \displaystyle \ln \left| x + \sqrt{1-x^2} \right| \arcsin x ln∣∣∣∣​x+1−x2​∣∣∣∣​arcsinx
  • \ln \left| x + \sqrt{1-x^2} \right|ln∣∣∣​x+1−x2​∣∣∣​
  • \displaystyle \frac{1}{\sqrt{1-x^2}}\ln \left| x + \sqrt{1-x^2} \right|1−x2​1​ln∣∣∣∣​x+1−x2​∣∣∣∣​
  • \ln \arcsin xlnarcsinx
  • \arcsin\left( \ln \left| x + \sqrt{1-x^2} \right| \right)arcsin(ln∣∣∣​x+1−x2​∣∣∣​)
  • 0

Q2. \displaystyle \frac{d}{dx} \int_{t=\sin x}^{\tan x} e^{-t^2} \, dt =dxd​∫t=sinxtanx​e−t2dt=

  • \displaystyle \int_{2\sin x\, cos x}^{2 \tan x\, \sec^2 x} e^{-t^2} \, dt∫2sinxcosx2tanxsec2x​e−t2dt
  • e^{-4\tan^2 x \sec^4 x} – e^{-4\sin^2 x \cos^2 x}e−4tan2xsec4x−e−4sin2xcos2x
  • 00
  • e^{-\tan^2 x}\sec^2 x – e^{-\sin^2 x}\cos xe−tan2xsec2x−e−sin2xcosx
  • 2e^{-\tan^2 x}\tan x \sec^2 x – 2e^{-\sin^2 x}\sin x \cos x2e−tan2xtanxsec2x−2e−sin2xsinxcosx
  • 2(\tan x \sec^2 x – \sin x \cos x)e^{-t^2}2(tanxsec2x−sinxcosx)e−t2

Q3. Which of the following is the leading order term in the Taylor series about x=0x=0 of

f(x) = \int_{0}^x\ln(\cosh(t)) dtf(x)=∫0x​ln(cosh(t))dt

Hint: yes, there’s more than one way to do this problem… Try using the F.T.I.C. to compute the derivatives.

  • \displaystyle \frac{x^2}{6}6x2​
  • \displaystyle \frac{x^2}{2}2x2​
  • \displaystyle -\frac{x^5}{60}−60x5​
  • \displaystyle \frac{x^3}{2}2x3​
  • \displaystyle \frac{x^3}{6}6x3​
  • \displaystyle -\frac{x^3}{18}−18x3​

Q4. We usually use Riemann sums to approximate integrals, but we can go the other way, too, using an antiderivative to approximate a sum. Using only your head (no paper, no calculator), tell me which of the following is the best estimate for

\sum_{n=0}^{100} n^3n=0∑100​n3

  • 2.5\times 10^72.5×107
  • 2.5\times 10^82.5×108
  • 6.6\times 10^76.6×107
  • 1.0\times 10^71.0×107
  • 3.3\times 10^73.3×107
  • 1.0\times 10^81.0×108

Main Quiz 02

Q1. \displaystyle \int_{x=-1}^1 \frac{dx}{1+x^2} =∫x=−11​1+x2dx​=

  • \displaystyle \frac{\pi}{4}4π​
  • \piπ
  • 2\pi2π
  • \displaystyle \frac{\pi}{2}2π​
  • 00
  • \displaystyle \frac{\pi}{3}3π​

Q2. \displaystyle \int_{x=0}^{3} 5x \sqrt{x+1} \, dx =∫x=03​5xx+1​dx=

  • \displaystyle \frac{116}{3}3116​
  • 3030
  • 12\sqrt{3}123​
  • 00
  • -12\sqrt{3}−123​
  • \displaystyle -\frac{116}{3}−3116​

Q3. \displaystyle \int_{x=-\pi}^\pi \frac{d}{dx} ( x\cos x ) \, dx =∫x=−ππ​dxd​(xcosx)dx=

  • x\cos x + Cxcosx+C
  • 00
  • 2\pi2π
  • \cos x – x\sin x + Ccosx−xsinx+C
  • x\cos xxcosx
  • -2\pi−2π

Q4. \displaystyle \frac{d}{dx} \int_{x=-\pi}^\pi x\cos x \, dx =dxd​∫x=−ππ​xcosxdx=

  • \cos x – x\sin x + Ccosx−xsinx+C
  • 2\pi2π
  • x\cos x + Cxcosx+C
  • x\cos xxcosx
  • 00
  • -2\pi−2π

Q5. \displaystyle \frac{d}{dx} \int_{t=0}^x \cos t \, dt =dxd​∫t=0x​costdt=

  • \sin xsinx
  • \sin x – 1sinx−1
  • 1 – \sin x1−sinx
  • 1 – \cos x1−cosx
  • \cos x – 1cosx−1
  • \cos xcosx

Week 04:Calculus: Single Variable Part 3 – Integration Coursera Quiz Answers

Main Quiz 01

Q1. Recall that we say that an integral is improper if:

  • the integrand blows up somewhere inside or on the boundary of the domain of integration,
  • the domain of integration is not bounded, or
  • both of the above.

You could also say that an integral that is not improper is proper (although this term is not used much).

Among the integrals below, select all of the improper ones and none of the proper ones.

  • \displaystyle \int_{\theta=-\pi/4}^{\pi/4} \sec \theta \, d\theta∫θ=−π/4π/4​secθdθ
  • \displaystyle \int_{x=0}^1 \frac{dx}{\sqrt{x} – 10}∫x=01​x​−10dx​
  • \displaystyle \int_{x=0}^1 \frac{dx}{x-1}∫x=01​x−1dx​
  • \displaystyle \int_{x=-1}^1 \ln(1+x^2) \, dx∫x=−11​ln(1+x2)dx
  • \displaystyle \int_{t=0}^{+\infty} t^{x-1} e^{-t} \, dt∫t=0+∞​tx−1e−tdt
  • \displaystyle \int_{x=0}^1 \frac{dx}{x^2}∫x=01​x2dx​
  • \displaystyle \int_{x=1}^{2} \sqrt{2-x} \, dx∫x=12​2−x​dx
  • \displaystyle \int_{\theta=\pi/4}^{3\pi/4} \sec\theta \, d\theta∫θ=π/43π/4​secθdθ
  • \displaystyle \int_{x=-\infty}^{+\infty} e^{-x^2/2} \, dx∫x=−∞+∞​e−x2/2dx
  • \displaystyle \int_{x=-1,000,000}^{1,000,000} e^{-x^2/2} \, dx∫x=−1,000,0001,000,000​e−x2/2dx

Q2. Consider the integral

\int_{x=0}^1 \frac{e^{-x}}{x} \, dx∫x=01​xe−x​dx

It is improper because its integrand blows up at x=0x=0. Using your knowledge of Taylor series, determine the leading order behavior of the integrand near x=0x=0 and decide whether the integral converges or diverges.

Note: observe that you do not need to find an antiderivative in order to determine whether an improper integral converges or diverges!

  • \displaystyle \frac{e^{-x}}{x} = 1 + O(x)xe−x​=1+O(x), so the integral converges.
  • \displaystyle \frac{e^{-x}}{x} = 1 + O(x)xe−x​=1+O(x), so the integral diverges.
  • \displaystyle \frac{e^{-x}}{x} = e^{-x} + O(xe^{-x})xe−x​=e−x+O(xe−x), so the integral converges.
  • \displaystyle \frac{e^{-x}}{x} = \frac{1}{x} + O(1)xe−x​=x1​+O(1), so the integral diverges.
  • \displaystyle \frac{e^{-x}}{x} = \frac{1}{x} + O(1)xe−x​=x1​+O(1), so the integral converges.
  • \displaystyle \frac{e^{-x}}{x} = e^{-x} + O(xe^{-x})xe−x​=e−x+O(xe−x), so the integral diverges.

Q3. By analyzing the leading order behavior of the integrand near x=0x=0, decide whether the following integral converges or diverges.

\int_{x=0}^1 \frac{\cos^2 x}{\sqrt{x}} \, dx∫x=01​x​cos2x​dx

  • \displaystyle \frac{\cos^2 x}{\sqrt{x}} = 1 + O(x^4)x​cos2x​=1+O(x4), so the integral diverges.
  • \displaystyle \frac{\cos^2 x}{\sqrt{x}} = \sqrt{x} + O(x^{3/2})x​cos2x​=x​+O(x3/2), so the integral converges.
  • \displaystyle \frac{\cos^2 x}{\sqrt{x}} = 1 + O(x^{4})x​cos2x​=1+O(x4), so the integral converges.
  • \displaystyle \frac{\cos^2 x}{\sqrt{x}} = \frac{1}{\sqrt{x}} + O(x^{3/2})x​cos2x​=x​1​+O(x3/2), so the integral converges.
  • \displaystyle \frac{\cos^2 x}{\sqrt{x}} = \frac{1}{\sqrt{x}} + O(x^{3/2})x​cos2x​=x​1​+O(x3/2), so the integral diverges.
  • \displaystyle \frac{\cos^2 x}{\sqrt{x}} = \sqrt{x} + O(x^{3/2})x​cos2x​=x​+O(x3/2), so the integral diverges.

Q4. Calculate

I = \int_{x=1}^2 \frac{dx}{\sqrt{x – 1}}I=∫x=12​x−1​dx​

Hint: remember from the Lecture that improper integrals are calculated as limits. In this case,

I = \lim_{T \to 1^+} \int_{x=T}^2 \frac{dx}{\sqrt{x – 1}}I=T→1+lim​∫x=T2​x−1​dx​

  • I = 1I=1
  • \displaystyle I = \frac{\sqrt{2}}{2} – 1I=22​​−1
  • I = -2I=−2
  • The integral diverges.
  • I = 2I=2
  • I = \sqrt{2} – 1I=2​−1

Q5. Calculate

I = \int_{x=0}^4 \frac{2 \, dx}{\sqrt{16 – x^2}}I=∫x=04​16−x2​2dx​

  • I = 0I=0
  • \displaystyle I = -\frac{\pi}{2}I=−2π​
  • I = -\piI=−π
  • I = \piI=π
  • \displaystyle I = \frac{\pi}{2}I=2π​
  • The integral diverges

Q6. Determine the leading order term of the integrand as x \to +\inftyx→+∞ to decide whether the following integral converges or diverges.

\int_{x=1}^{+\infty} \frac{\sqrt[3]{x + 3}}{x^3} \, dx∫x=1+∞​x33x+3​​dx

  • \displaystyle \frac{\sqrt[3]{x + 3}}{x^3} = \frac{1}{x^3} + O\left(\frac{1}{x^4}\right)x33x+3​​=x31​+O(x41​), so the integral diverges.
  • \displaystyle \frac{\sqrt[3]{x + 3}}{x^3} = \frac{1}{x^{10/3}} + O\left(\frac{1}{x^{13/3}}\right)x33x+3​​=x10/31​+O(x13/31​), so the integral diverges.
  • \displaystyle \frac{\sqrt[3]{x + 3}}{x^3} = \frac{1}{x^{8/3}} + O\left(\frac{1}{x^{11/3}}\right)x33x+3​​=x8/31​+O(x11/31​), so the integral diverges.
  • \displaystyle \frac{\sqrt[3]{x + 3}}{x^3} = \frac{1}{x^3} + O\left(\frac{1}{x^4}\right)x33x+3​​=x31​+O(x41​), so the integral converges.
  • \displaystyle \frac{\sqrt[3]{x + 3}}{x^3} = \frac{1}{x^{8/3}} + O\left(\frac{1}{x^{11/3}}\right)x33x+3​​=x8/31​+O(x11/31​), so the integral converges.
  • \displaystyle \frac{\sqrt[3]{x + 3}}{x^3} = \frac{1}{x^{10/3}} + O\left(\frac{1}{x^{13/3}}\right)x33x+3​​=x10/31​+O(x13/31​), so the integral converges.

Q7. Does the integral converge or diverge? \int_{x=1}^{+\infty} \frac{1-5^{-x}}{x} \, dx∫x=1+∞​x1−5−x​dx

  • The leading order term in the integrand as x \to +\inftyx→+∞ is 5^{-x}5−x, so the integral converges.
  • The leading order term in the integrand as x \to +\inftyx→+∞ is 11, so the integral diverges.
  • The leading order term in the integrand as x \to +\inftyx→+∞ is 5^{-x}5−x, so the integral diverges.
  • The leading order term in the integrand as x \to +\inftyx→+∞ is \displaystyle \frac{1}{x}x1​, so the integral converges.
  • The leading order term in the integrand as x \to +\inftyx→+∞ is 11, so the integral converges.
  • The leading order term in the integrand as x \to +\inftyx→+∞ is \displaystyle \frac{1}{x}x1​, so the integral diverges.

Q8. Calculate

I = \int_{x=0}^{+\infty} e^{-x} \sin x \, dxI=∫x=0+∞​e−xsinxdx

Hint: again, remember that this integral is defined as the limit

I = \lim_{T \to +\infty} \int_{x=0}^T e^{-x} \sin x \, dxI=T→+∞lim​∫x=0T​e−xsinxdx

  • I = 0I=0
  • \displaystyle I = -\frac{1}{2}I=−21​
  • I = -1I=−1
  • \displaystyle I = \frac{1}{2}I=21​
  • I = 1I=1
  • The integral diverges

Practice Quiz 01

Q1`. Look at the following integral:

\int_{x=1}^{+\infty} \frac{1}{\sqrt{x^3 -1} } \, dx∫x=1+∞​x3−1​1​dx

The integrand blows up at x=1x=1, and so it is an improper integral. But it is also improper because the domain of integration is unbounded!

Analyze the behavior of the integrand both near x=1x=1 and as x \to +\inftyx→+∞ to decide whether the integral converges or diverges. From the statements below, choose all the correct ones and none of the incorrect ones.

  • The integral diverges.
  • The integral converges.
  • Near x=1x=1, the integrand is \displaystyle \frac{1}{\sqrt{3(x-1)}} + O\left(\sqrt{x-1}\right)3(x−1)​1​+O(x−1​).
  • As x \to +\inftyx→+∞, the integrand is \displaystyle \frac{1}{x^{3/2}} + O\left(\frac{1}{x^{9/2}}\right)x3/21​+O(x9/21​).
  • As x \to +\inftyx→+∞, the integrand is \displaystyle \frac{1}{x^{1/3}} + O(x^{2/3})x1/31​+O(x2/3).
  • Near x=1x=1, the integrand is \displaystyle \frac{1}{(x-1)^{3/2}} + O\left(\frac{1}{(x-1)^{1/2}}\right)(x−1)3/21​+O((x−1)1/21​).

Q2. Consider the following two integrals:

I_1 = \int_{x=2}^{+\infty} \frac{dx}{\sqrt{x^3 – 8}}, \qquad I_2 = \int_{x=2}^{+\infty} \frac{1}{\sqrt{(x-2)^3}} \, dxI1​=∫x=2+∞​x3−8​dx​,I2​=∫x=2+∞​(x−2)3​1​dx

They look very much alike. In fact, both of their integrands blow up at x=2x=2, and the domain of integration —the same one in both cases— is unbounded. But which of the following statements is true?

  • Both I_1I1​ and I_2I2​ diverge.
  • Both I_1I1​ and I_2I2​ converge.
  • I_2I2​ converges, but I_1I1​ diverges.
  • I_1I1​ converges, but I_2I2​ diverges.

Q3. Until now we have used asymptotic analysis to relate an improper integral to a pp-integral. But sometimes the leading order term is not a power of xx…

Identify the leading order term as x \to +\inftyx→+∞ of the integrand of

\int_{x=1}^{+\infty} \frac{1}{\sinh x} \, dx∫x=1+∞​sinhx1​dx

Use this information to decide whether it converges or diverges.

  • \displaystyle \frac{1}{\sinh x} = \frac{e^{-x}}{x} + O(xe^{-x})sinhx1​=xe−x​+O(xe−x), so the integral diverges.
  • \displaystyle \frac{1}{\sinh x} = \frac{e^{-x}}{x} + O(xe^{-x})sinhx1​=xe−x​+O(xe−x), so the integral converges.
  • \displaystyle \frac{1}{\sinh x} = e^x + O(e^{3x})sinhx1​=ex+O(e3x), so the integral converges.
  • \displaystyle \frac{1}{\sinh x} = e^x + O(e^{3x})sinhx1​=ex+O(e3x), so the integral diverges.
  • \displaystyle \frac{1}{\sinh x} = 2e^{-x} + O(e^{-3x})sinhx1​=2e−x+O(e−3x), so the integral converges.
  • \displaystyle \frac{1}{\sinh x} = 2e^{-x} + O(e^{-3x})sinhx1​=2e−x+O(e−3x), so the integral diverges.

Q4. For p \geq 0p≥0 an integer, consider the following integral:I_p = \int_{x=1}^{+\infty} \frac{dx}{\ln^p x}Ip​=∫x=1+∞​lnpxdx​ Which of the following statements is true?

Two hints:

  1. Think about the growth of \ln^p xlnpx as x \to +\inftyx→+∞. Can you compare it to the growth of x^qxq for some value of qq ?
  2. Recall from Lecture 25 that if g(x) \geq f(x)g(x)≥f(x) for every x \in [a,b]x∈[a,b], then\int_{x=a}^b g(x) \,dx \geq \int_{x=a}^b f(x) \,dx∫x=ab​g(x)dx≥∫x=ab​f(x)dx This is also true if the domain of integration is unbounded! Now, from the comparison between the orders of growth above, what can you deduce?
  • I_pIp​ converges for any value of pp.
  • I_pIp​ converges for p \gt 1p>1 and diverges for p \leq 1p≤1.
  • I_pIp​ converges for p \leq 1p≤1 and diverges for p \gt 1p>1.
  • I_pIp​ diverges for any value of pp.
  • I_pIp​ converges for p \geq 1p≥1 and diverges for p=0p=0.

Q5. We have alluded to the fact that the integral of e^{-t^2}e−t2 is difficult. Let’s say I told you that

\int_{0}^\infty e^{-t^2}dt = \frac{\sqrt{\pi}}{2} .∫0∞​e−t2dt=2π​​.

Use this fact to compute \left(-\frac{1}{2}\right)!(−21​)! (that is, negative one-half, factorial).

Hint: use the \GammaΓ-function and a substitution.

  • \displaystyle\frac{\pi}{2}2π​
  • \pi^2π2
  • \displaystyle\frac{\pi^2}{2}2π2​
  • \piπ
  • \sqrt{\pi}π​
  • \displaystyle\frac{\sqrt{\pi}}{2}2π​​

Main Quiz 02

Q1. \displaystyle \int \sin ^2 x \cos ^2 x \, dx =∫sin2xcos2xdx=

Hint: you may (or may not) need to use any of the following reduction formulae:

\int \cos^n x \, dx = \frac{\cos^{n-1}x \sin x}{n} + \frac{n-1}{n} \int \cos^{n-2}x \, dx∫cosnxdx=ncosn−1xsinx​+nn−1​∫cosn−2xdx

\int \sin^n x \, dx = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n} \int \sin^{n-2}x \, dx∫sinnxdx=−nsinn−1xcosx​+nn−1​∫sinn−2xdx

More than one answer may be correct.

  • \displaystyle \frac{1}{8} x + \frac{1}{16} \sin 2x – \frac{1}{4} \sin x \cos^3 x + C81​x+161​sin2x−41​sinxcos3x+C
  • \displaystyle \frac{1}{8} x + \frac{1}{32} \cos 4x + C81​x+321​cos4x+C
  • \displaystyle \frac{1}{8} x + \frac{1}{16} \cos 2x – \frac{1}{4} \sin x \cos^3 x + C81​x+161​cos2x−41​sinxcos3x+C
  • \displaystyle \frac{1}{8} x – \frac{1}{16} \cos 2x + \frac{1}{4} \sin^3 x \cos x + C81​x−161​cos2x+41​sin3xcosx+C
  • \displaystyle \frac{1}{8} x – \frac{1}{16} \sin 2x + \frac{1}{4} \sin^3 x \cos x + C81​x−161​sin2x+41​sin3xcosx+C
  • \displaystyle \frac{1}{8} x – \frac{1}{32} \sin 4x + C81​x−321​sin4x+C

Q2. \displaystyle \int \sin ^3 \frac{x}{2} \cos ^3 \frac{x}{2} \, dx =∫sin32x​cos32x​dx=

(More than one answer may be correct.)

  • \displaystyle -\frac{1}{3} \sin^6 \frac{x}{2} + \frac{1}{2} \sin^4 \frac{x}{2} + C−31​sin62x​+21​sin42x​+C
  • \displaystyle \frac{1}{3} \cos^6 \frac{x}{2} -\frac{1}{2} \cos^4 \frac{x}{2} + C31​cos62x​−21​cos42x​+C
  • \displaystyle \frac{1}{24} \cos^3 x -\frac{1}{8} \cos x + C241​cos3x−81​cosx+C
  • \displaystyle \frac{1}{6} \sin^6 \frac{x}{2} – \frac{1}{4} \sin^4 \frac{x}{2} + C61​sin62x​−41​sin42x​+C
  • \displaystyle \frac{1}{96} \big( \cos x – 9\cos 3x \big) + C961​(cosx−9cos3x)+C
  • \displaystyle -2\sin^5 \frac{x}{2} + 2 \sin^3 \frac{x}{2} + C−2sin52x​+2sin32x​+C

Q3. \displaystyle \int \frac{x^3\, dx}{\sqrt{9-x^2}} =∫9−x2​x3dx​=

Hint: use a trigonometric substitution to obtain a trigonometric integral.

  • \displaystyle – \bigg( 6 + \frac{x^2}{3} \bigg) \sqrt{9-x^2} + C−(6+3x2​)9−x2​+C
  • \displaystyle -3\sqrt{9-x^2} – \frac{1}{81} (9-x^2)^{3/2} + C−39−x2​−811​(9−x2)3/2+C
  • \displaystyle \frac{1}{6} (9-x^2)^{1/2} + C61​(9−x2)1/2+C
  • 3 (9-x^2)^{3/2} + C3(9−x2)3/2+C
  • \displaystyle \bigg( \frac{1}{3} + 9x^2 \bigg) \sqrt{9-x^2} + C(31​+9x2)9−x2​+C
  • \displaystyle 9\sqrt{9-x^2} – \frac{1}{3} (9-x^2)^{3/2} + C99−x2​−31​(9−x2)3/2+C

Q4. \displaystyle \int 5\tan ^5 x \sec ^3 x \, dx =∫5tan5xsec3xdx=

  • \displaystyle\frac{5}{6} \tan^6 x + \frac{5}{8} \tan^8 x + C65​tan6x+85​tan8x+C
  • \displaystyle\frac{5}{7} \sec^{14} x – 2\sec^{10} x + \frac{5}{3} \sec^6 x + C75​sec14x−2sec10x+35​sec6x+C
  • \displaystyle \sec^5 x – \frac{10}{3} \sec^3 x + 5\sec x + Csec5x−310​sec3x+5secx+C
  • \displaystyle\frac{4}{5} \sec^5 x \tan^5 x – \frac{4}{3} \sec^3 x \tan^3 x + C54​sec5xtan5x−34​sec3xtan3x+C
  • \displaystyle \frac{5}{7} \tan^7 x + \tan^5 x + C75​tan7x+tan5x+C
  • \displaystyle \frac{5}{7} \sec^7 x – 2 \sec^5 x + \frac{5}{3} \sec^3 x + C75​sec7x−2sec5x+35​sec3x+C

Q5. \displaystyle \int 7\tan ^4 x \sec ^4 x \, dx =∫7tan4xsec4xdx=

  • \displaystyle \frac{1}{7} \tan^7 x + \frac{1}{5} \tan^5 x + C71​tan7x+51​tan5x+C
  • \displaystyle \frac{7}{5} \tan^5 x + \frac{7}{3} \tan^3 x + C57​tan5x+37​tan3x+C
  • \displaystyle \sec^{7} x + \frac{7}{5} \sec^{5} x + Csec7x+57​sec5x+C
  • \displaystyle \tan^7 x + \frac{7}{5} \tan^5 x + Ctan7x+57​tan5x+C
  • \displaystyle \frac{1}{2} \sec^{14} x + \frac{7}{10} \sec^{10} x + C21​sec14x+107​sec10x+C
  • \displaystyle \tan^7 x – \frac{7}{5} \tan^5 x + Ctan7x−57​tan5x+C

Q6. Assuming 0 \lt x \lt \pi0<x<π, compute

\displaystyle \int \cot x \sqrt{1 – \cos 2x} \, dx =∫cotx1−cos2x​dx=

Hint: remember the double-angle formula

\cos 2x = \cos^2 x – \sin^2 xcos2x=cos2x−sin2x

  • \cos^2 x + Ccos2x+C
  • \sqrt{2} \cos x + C2​cosx+C
  • \displaystyle \frac{x}{2} + \frac{ \cos 2x}{2} + C2x​+2cos2x​+C
  • \sqrt{2} \sin x + C2​sinx+C
  • \cos x + Ccosx+C
  • \sin x + Csinx+C

Main Quiz 03

Q1. \displaystyle \int \frac{(\arcsin x)^2}{\sqrt{1-x^2}}\, dx =∫1−x2​(arcsinx)2​dx=

  • \displaystyle \frac{1}{3} \sin^3 x + C31​sin3x+C
  • \displaystyle \frac{1}{3} \left( \arcsin\sqrt{1-x^2} \right)^3 + C31​(arcsin1−x2​)3+C
  • \displaystyle \frac{1}{3} \sqrt{1-(\arcsin x)^3} + C31​1−(arcsinx)3​+C
  • \displaystyle \frac{2}{3} \sqrt{1-\sin^3 x} + C32​1−sin3x​+C
  • \displaystyle \sqrt{\frac{1}{3}(\arcsin x)^3} + C31​(arcsinx)3​+C
  • \displaystyle \frac{(\arcsin x)^3}{3\sqrt{1-x^2}} + C31−x2​(arcsinx)3​+C
  • \displaystyle \frac{1}{3} (\arcsin x)^3 + C31​(arcsinx)3+C
  • \displaystyle \frac{x^3}{3} + C3x3​+C

Q2. \displaystyle \int \frac{dx}{x^2-x-6} =∫x2−x−6dx​=

  • \displaystyle \frac{1}{3} \ln\left| \frac{x-3}{x+2} \right| + C31​ln∣∣∣∣∣​x+2x−3​∣∣∣∣∣​+C
  • \displaystyle -\frac{1}{3} \ln\left| \frac{x-3}{x+2} \right| + C−31​ln∣∣∣∣∣​x+2x−3​∣∣∣∣∣​+C
  • \displaystyle -\frac{1}{3} \ln\left| \frac{x+3}{x-2} \right| + C−31​ln∣∣∣∣∣​x−2x+3​∣∣∣∣∣​+C
  • \displaystyle \frac{1}{5} \ln\left| \frac{x-3}{x+2} \right| + C51​ln∣∣∣∣∣​x+2x−3​∣∣∣∣∣​+C
  • \displaystyle -\frac{1}{5} \ln\left| \frac{x-3}{x+2} \right| + C−51​ln∣∣∣∣∣​x+2x−3​∣∣∣∣∣​+C
  • \displaystyle -\frac{1}{5} \ln\left| \frac{x+3}{x-2} \right| + C−51​ln∣∣∣∣∣​x−2x+3​∣∣∣∣∣​+C
  • \displaystyle \frac{1}{3} \ln\left| \frac{x+3}{x-2} \right| + C31​ln∣∣∣∣∣​x−2x+3​∣∣∣∣∣​+C
  • \displaystyle \frac{1}{5} \ln\left| \frac{x+3}{x-2} \right| + C51​ln∣∣∣∣∣​x−2x+3​∣∣∣∣∣​+C

Q3. Exactly half of the following integrals converge: which are they? In order to receive full credit for this problem, you must select all the integrals that converge and none of those that diverge.

  • \displaystyle \int_{x=0}^1 \frac{\cos x}{x^2}dx∫x=01​x2cosx​dx
  • \displaystyle \int_{x=0}^3 \frac{dx}{x-1}∫x=03​x−1dx​
  • \displaystyle \int_{x=0}^1 \frac{\sin(\pi x)}{x^{3/2}} dx∫x=01​x3/2sin(πx)​dx
  • \displaystyle \int_{x=1}^2 \frac{2}{\sqrt[3]{x-1}} dx∫x=12​3x−1​2​dx
  • \displaystyle \int_{x=0}^{+\infty} \frac{2x + 3}{x^2 + 3x + 6}\, dx∫x=0+∞​x2+3x+62x+3​dx
  • \displaystyle \int_{x=2}^{+\infty} \frac{e^{-x}}{x}dx∫x=2+∞​xe−x​dx
  • \displaystyle \int_{x=2}^{+\infty} \frac{dx}{x^{1/2}(x-1)^{3/2}}∫x=2+∞​x1/2(x−1)3/2dx​
  • \displaystyle \int_{x=2}^{+\infty} \frac{x}{1+x^2}dx∫x=2+∞​1+x2x​dx

Q4. \displaystyle \int_{x=0}^{\pi/2} x \sin 2x \,dx =∫x=0π/2​xsin2xdx=

  • \displaystyle \frac{1}{4}41​
  • \displaystyle -\frac{\pi}{4}−4π​
  • \displaystyle -\frac{1}{4}−41​
  • \displaystyle \frac{\pi}{2}2π​
  • \displaystyle -\frac{1}{2}−21​
  • \displaystyle -\frac{\pi}{2}−2π​
  • \displaystyle \frac{\pi}{4}4π​
  • \displaystyle \frac{1}{2}21​

Q5. Compute \displaystyle \int \frac{dx}{x^2-2x+10}∫x2−2x+10dx​. Hint: partial fractions doesn’t seem to work on this problem, since you don’t have real roots for the denominator…is there another technique you could use?

  • \displaystyle \frac{1}{3} \arctan\frac{x+1}{5} + C31​arctan5x+1​+C
  • \displaystyle \frac{1}{2} \arctan\frac{x+1}{2} + C21​arctan2x+1​+C
  • \displaystyle \frac{1}{3} \arctan\frac{x+1}{3} + C31​arctan3x+1​+C
  • \displaystyle \frac{1}{3} \arctan\frac{x-1}{5} + C31​arctan5x−1​+C
  • \displaystyle \frac{1}{2} \arctan\frac{x-1}{4} + C21​arctan4x−1​+C
  • \displaystyle \frac{1}{2} \arctan\frac{x+1}{4} + C21​arctan4x+1​+C
  • \displaystyle \frac{1}{2} \arctan\frac{x-1}{2} + C21​arctan2x−1​+C
  • \displaystyle \frac{1}{3} \arctan\frac{x-1}{3} + C31​arctan3x−1​+C

Q6. Solve the following separable differential equation \displaystyle \frac{dx}{dt} = e^{t-x}dtdx​=et−x.

  • \displaystyle x = -\ln\left(C-e^t\right)x=−ln(C−et)
  • \displaystyle x = \left( \frac{t^2}{2} + C \right) e^{-2t}x=(2t2​+C)e−2t
  • \displaystyle x = \ln\left(Ce^{-2t}\right)x=ln(Ce−2t)
  • \displaystyle x = \frac{t}{2} – \frac{1}{4} + Cx=2t​−41​+C
  • \displaystyle x = \frac{t^2}{4} + \frac{1}{2} + Ce^{-2t}x=4t2​+21​+Ce−2t
  • \displaystyle x = -\frac{t}{2} – \frac{1}{4} + Ce^{2t}x=−2t​−41​+Ce2t
  • \displaystyle x = \ln\left(e^t+C\right)x=ln(et+C)
  • \displaystyle x =t+ \ln Cx=t+lnC

Q7. Compute \displaystyle \int \cos^3 4x \,dx∫cos34xdx .

  • \displaystyle \frac{1}{16} \cos^4 4x + C161​cos44x+C
  • \displaystyle \cos 4x + \frac{1}{4}\cos^4 4x + Ccos4x+41​cos44x+C
  • \displaystyle \frac{1}{4} \cos^4 4x + C41​cos44x+C
  • \displaystyle \frac{1}{4}\cos 4x + \frac{1}{12}\cos^4 4x + C41​cos4x+121​cos44x+C
  • \displaystyle \sin 4x – \frac{1}{3}\sin^3 4x + Csin4x−31​sin34x+C
  • \displaystyle \frac{1}{4}\sin 4x – \frac{1}{12}\sin^3 4x + C41​sin4x−121​sin34x+C
  • 12\cos^2 4x + C12cos24x+C
  • \displaystyle \frac{1}{4}\sin 4x + \frac{1}{12}\sin^3 4x + C41​sin4x+121​sin34x+C

Q8. \displaystyle \left.\frac{d}{dx}\right|_{x=\pi} \int_{t=0}^{x} \frac{\cos 3t}{\sqrt{1+t}} \,dt =dxd​∣∣∣∣∣​x=π​∫t=0x​1+t​cos3t​dt=

  • \displaystyle \frac{\pi}{\sqrt{1+\pi}}1+π​π​
  • 00
  • -\pi−π
  • \displaystyle -\frac{1}{\sqrt{1+\pi}}−1+π​1​
  • \displaystyle \int_0^{\pi} \frac{d}{dt}\!\left( \frac{\cos 3t}{\sqrt{1+t}} \right) dt∫0π​dtd​(1+t​cos3t​)dt
  • \displaystyle -1−1
  • \displaystyle \frac{\cos(3\ln \pi)}{\pi\sqrt{1+\ln \pi}}π1+lnπ​cos(3lnπ)​
  • \displaystyle \frac{1}{\sqrt{1+\pi}}1+π​1​

Q9. Which of the following is the integrating factor, II, used to solve the linear differential equation

e^{3t}\frac{dx}{dt}=2-e^txe3tdtdx​=2−etx

  • I = e^{e^{t}}I=eet, that is, I= {\rm exp}\left( e^{t}\right)I=exp(et)
  • I = 2e^{3t}I=2e3t, that is, I= 2\,{\rm exp }\left( 3t\right)I=2exp(3t)
  • I = e^{\frac{1}{2}e^{2t}}I=e21​e2t, that is, I= {\rm exp}\left(\frac{1}{2}e^{2t}\right)I=exp(21​e2t)
  • I = e^{-e^{-t}}I=e−e−t, that is, I= {\rm exp}\left( -e^{-t}\right)I=exp(−e−t)
  • I = 2e^{-3t}I=2e−3t, that is, I= 2\,{\rm exp }\left( -3t\right)I=2exp(−3t)
  • I = e^{\frac{1}{3}e^{3t}}I=e31​e3t, that is, I= {\rm exp}\left( \frac{1}{3}e^{3t}\right)I=exp(31​e3t)
  • I = e^{-\frac{1}{3}e^{-3t}}I=e−31​e−3t, that is, I= {\rm exp}\left( -\frac{1}{3}e^{-3t}\right)I=exp(−31​e−3t)
  • I = e^{-\frac{1}{2}e^{-2t}}I=e−21​e−2t, that is, I= {\rm exp}\left( -\frac{1}{2}e^{-2t}\right)I=exp(−21​e−2t)

Q10. The size z(t)z(t) of a hailstone evolves according to the differential equation

\frac{dz}{dt} = A \sqrt{z} – B\sqrt{z^3}dtdz​=Az​−Bz3​

where AA and BB are positive constants. Without solving the differential equation, determine the limiting size \displaystyle \lim_{t\to+\infty}z(t)t→+∞lim​z(t) in the case where z(0)=1z(0)=1. Hint: think in terms of equilibria…

  • \displaystyle\frac{\sqrt{A}}{B}BA​​
  • \displaystyle-\frac{A}{B}−BA​
  • \displaystyle\sqrt{\frac{A}{B}}BA​​
  • \displaystyle\frac{A}{B}BA​
  • \displaystyle-\frac{B}{A}−AB​
  • \displaystyle\frac{B}{A}AB​
  • \displaystyle\frac{A}{\sqrt{B}}B​A​
  • \displaystyle\sqrt{\frac{B}{A}}AB

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  767. Espectro de vibracion
    Dispositivos de calibracion: fundamental para el operacion suave y efectivo de las dispositivos.

    En el campo de la tecnologia moderna, donde la rendimiento y la estabilidad del equipo son de gran trascendencia, los sistemas de equilibrado juegan un tarea esencial. Estos equipos adaptados estan concebidos para equilibrar y fijar partes rotativas, ya sea en equipamiento industrial, transportes de movilidad o incluso en electrodomesticos hogarenos.

    Para los especialistas en reparacion de dispositivos y los ingenieros, utilizar con equipos de balanceo es fundamental para promover el funcionamiento fluido y confiable de cualquier sistema giratorio. Gracias a estas alternativas avanzadas avanzadas, es posible limitar notablemente las vibraciones, el sonido y la presion sobre los rodamientos, mejorando la vida util de partes costosos.

    Asimismo relevante es el papel que tienen los equipos de ajuste en la asistencia al comprador. El apoyo especializado y el conservacion permanente utilizando estos dispositivos facilitan ofrecer servicios de alta nivel, mejorando la contento de los compradores.

    Para los duenos de proyectos, la aporte en equipos de balanceo y dispositivos puede ser importante para incrementar la efectividad y rendimiento de sus equipos. Esto es particularmente significativo para los inversores que dirigen pequenas y medianas empresas, donde cada aspecto cuenta.

    Ademas, los aparatos de equilibrado tienen una amplia uso en el sector de la prevencion y el supervision de estandar. Posibilitan encontrar probables defectos, reduciendo intervenciones onerosas y averias a los dispositivos. Mas aun, los resultados obtenidos de estos dispositivos pueden emplearse para perfeccionar sistemas y aumentar la presencia en motores de consulta.

    Las campos de aplicacion de los sistemas de ajuste abarcan numerosas sectores, desde la fabricacion de vehiculos de dos ruedas hasta el supervision del medio ambiente. No afecta si se trata de importantes elaboraciones manufactureras o pequenos locales caseros, los equipos de balanceo son fundamentales para garantizar un funcionamiento eficiente y sin interrupciones.

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  816. Equilibrado
    Equipos de calibración: importante para el funcionamiento fluido y efectivo de las dispositivos.

    En el entorno de la innovación avanzada, donde la efectividad y la fiabilidad del sistema son de máxima importancia, los dispositivos de equilibrado desempeñan un función vital. Estos aparatos específicos están creados para ajustar y estabilizar componentes móviles, ya sea en herramientas de fábrica, transportes de desplazamiento o incluso en aparatos de uso diario.

    Para los especialistas en soporte de dispositivos y los profesionales, utilizar con aparatos de calibración es importante para promover el funcionamiento uniforme y seguro de cualquier mecanismo móvil. Gracias a estas soluciones avanzadas sofisticadas, es posible minimizar sustancialmente las oscilaciones, el ruido y la tensión sobre los rodamientos, prolongando la longevidad de partes valiosos.

    Asimismo trascendental es el función que juegan los sistemas de balanceo en la soporte al cliente. El apoyo experto y el conservación continuo utilizando estos sistemas posibilitan brindar soluciones de alta nivel, elevando la agrado de los usuarios.

    Para los propietarios de emprendimientos, la aporte en estaciones de equilibrado y sensores puede ser fundamental para aumentar la productividad y desempeño de sus aparatos. Esto es particularmente relevante para los emprendedores que administran modestas y modestas organizaciones, donde cada elemento es relevante.

    Además, los dispositivos de ajuste tienen una vasta implementación en el área de la fiabilidad y el monitoreo de excelencia. Facilitan identificar posibles errores, impidiendo intervenciones onerosas y averías a los sistemas. Incluso, los indicadores generados de estos aparatos pueden utilizarse para maximizar procedimientos y aumentar la presencia en plataformas de exploración.

    Las sectores de implementación de los aparatos de calibración abarcan diversas industrias, desde la manufactura de ciclos hasta el supervisión ecológico. No afecta si se habla de importantes elaboraciones manufactureras o limitados espacios domésticos, los dispositivos de equilibrado son indispensables para asegurar un rendimiento productivo y sin riesgo de interrupciones.

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  978. Equilibrado de piezas
    El Equilibrado de Piezas: Clave para un Funcionamiento Eficiente

    ¿ Has percibido alguna vez temblores inusuales en un equipo industrial? ¿O sonidos fuera de lo común? Muchas veces, el problema está en algo tan básico como una falta de simetría en un elemento móvil. Y créeme, ignorarlo puede costarte caro .

    El equilibrado de piezas es un procedimiento clave en la producción y cuidado de equipos industriales como ejes, volantes, rotores y partes de motores eléctricos . Su objetivo es claro: impedir oscilaciones que, a la larga, puedan provocar desperfectos graves.

    ¿Por qué es tan importante equilibrar las piezas?
    Imagina que tu coche tiene una llanta mal nivelada . Al acelerar, empiezan las sacudidas, el timón vibra y resulta incómodo circular así. En maquinaria industrial ocurre algo similar, pero con consecuencias mucho más graves :

    Aumento del desgaste en soportes y baleros
    Sobrecalentamiento de partes críticas
    Riesgo de fallos mecánicos repentinos
    Paradas imprevistas que exigen arreglos costosos
    En resumen: si no se corrige a tiempo, una leve irregularidad puede transformarse en un problema grave .

    Métodos de equilibrado: cuál elegir
    No todos los casos son iguales. Dependiendo del tipo de pieza y su uso, se aplican distintas técnicas:

    Equilibrado dinámico
    Ideal para piezas que giran a alta velocidad, como rotores o ejes . Se realiza en máquinas especializadas que detectan el desequilibrio en dos o más planos . Es el método más preciso para garantizar un funcionamiento suave .
    Equilibrado estático
    Se usa principalmente en piezas como neumáticos, discos o volantes de inercia. Aquí solo se corrige el peso excesivo en una única dirección. Es ágil, práctico y efectivo para determinados sistemas.
    Corrección del desequilibrio: cómo se hace
    Taladrado selectivo: se quita peso en el punto sobrecargado
    Colocación de contrapesos: por ejemplo, en llantas o aros de volantes
    Ajuste de masas: común en cigüeñales y otros componentes críticos
    Equipos profesionales para detectar y corregir vibraciones
    Para hacer un diagnóstico certero, necesitas herramientas precisas. Hoy en día hay opciones económicas pero potentes, tales como:

    ✅ Balanset-1A — Tu asistente móvil para analizar y corregir oscilaciones

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  988. Equilibrado dinámico portátil:
    Respuesta inmediata sin mover equipos

    Imagina esto: tu rotor inicia con movimientos anormales, y cada minuto de inactividad afecta la productividad. ¿Desmontar la máquina y esperar días por un taller? Ni pensarlo. Con un equipo de equilibrado portátil, corriges directamente en el lugar en horas, sin alterar su posición.

    ¿Por qué un equilibrador móvil es como un “herramienta crítica” para máquinas rotativas?
    Fácil de transportar y altamente funcional, este dispositivo es la herramienta que todo técnico debería tener a mano. Con un poco de práctica, puedes:
    ✅ Prevenir averías mayores al detectar desbalances.
    ✅ Reducir interrupciones no planificadas.
    ✅ Trabajar en lugares remotos, desde plataformas petroleras hasta plantas eólicas.

    ¿Cuándo es ideal el equilibrado rápido?
    Siempre que puedas:
    – Acceder al rotor (eje, ventilador, turbina, etc.).
    – Ubicar dispositivos de medición sin inconvenientes.
    – Realizar ajustes de balance mediante cambios de carga.

    Casos típicos donde conviene usarlo:
    La máquina muestra movimientos irregulares o ruidos atípicos.
    No hay tiempo para desmontajes (operación prioritaria).
    El equipo es difícil de parar o caro de inmovilizar.
    Trabajas en zonas remotas sin infraestructura técnica.

    Ventajas clave vs. llamar a un técnico
    | Equipo portátil | Servicio externo |
    |—————-|——————|
    | ✔ Rápida intervención (sin demoras) | ❌ Retrasos por programación y transporte |
    | ✔ Monitoreo preventivo (evitas fallas mayores) | ❌ Solo se recurre ante fallos graves |
    | ✔ Reducción de costos operativos con uso continuo | ❌ Gastos periódicos por externalización |

    ¿Qué máquinas se pueden equilibrar?
    Cualquier sistema rotativo, como:
    – Turbinas de vapor/gas
    – Motores industriales
    – Ventiladores de alta potencia
    – Molinos y trituradoras
    – Hélices navales
    – Bombas centrífugas

    Requisito clave: acceso suficiente para medir y corregir el balance.

    Tecnología que simplifica el proceso
    Los equipos modernos incluyen:
    Aplicaciones didácticas (para usuarios nuevos o técnicos en formación).
    Evaluación continua (informes gráficos comprensibles).
    Autonomía prolongada (ideales para trabajo en campo).

    Ejemplo práctico:
    Un molino en una mina comenzó a vibrar peligrosamente. Con un equipo portátil, el técnico identificó el problema en menos de media hora. Lo corrigió añadiendo contrapesos y impidió una interrupción prolongada.

    ¿Por qué esta versión es más efectiva?
    – Estructura más dinámica: Organización visual facilita la comprensión.
    – Enfoque práctico: Ofrece aplicaciones tangibles del método.
    – Lenguaje persuasivo: Frases como “recurso vital” o “evitas fallas mayores” refuerzan el valor del servicio.
    – Detalles técnicos útiles: Se especifican requisitos y tecnologías modernas.

    ¿Necesitas ajustar el tono (más técnico) o añadir keywords específicas? ¡Aquí estoy para ayudarte! ️

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  991. ¿Movimientos irregulares en tu maquinaria? Servicio de balanceo dinámico en campo y venta de equipos.

    ¿Has detectado oscilaciones anómalas, zumbidos inesperados o deterioro prematuro en tus equipos? Estos son señales claras de que tu equipo industrial necesita un ajuste de precisión especializado.

    En lugar de desmontar y enviar tus máquinas a un taller, nuestros técnicos se desplazan a tu fábrica con equipos de última generación para resolver el problema sin detener tus procesos.

    Beneficios de nuestro servicio de equilibrado in situ
    ✔ Evitamos desarmados y transportes — Realizamos el servicio en tu locación.
    ✔ Evaluación detallada — Utilizamos tecnología avanzada para detectar la causa.
    ✔ Efectos al instante — Soluciones rápidas en cuestión de horas.
    ✔ Informe detallado — Certificamos el proceso con datos comparativos.
    ✔ Especialización en múltiples industrias — Solucionamos problemas en maquinaria pesada y liviana.

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  993. analizador de vibrasiones
    Balanceo móvil en campo:
    Soluciones rápidas sin desmontar máquinas

    Imagina esto: tu rotor inicia con movimientos anormales, y cada minuto de inactividad cuesta dinero. ¿Desmontar la máquina y esperar días por un taller? Olvídalo. Con un equipo de equilibrado portátil, solucionas el problema in situ en horas, sin mover la maquinaria.

    ¿Por qué un equilibrador móvil es como un “kit de supervivencia” para máquinas rotativas?
    Pequeño, versátil y eficaz, este dispositivo es la herramienta que todo técnico debería tener a mano. Con un poco de práctica, puedes:
    ✅ Prevenir averías mayores al detectar desbalances.
    ✅ Minimizar tiempos muertos y mantener la operación.
    ✅ Operar en zonas alejadas, ya sea en instalaciones marítimas o centrales solares.

    ¿Cuándo es ideal el equilibrado rápido?
    Siempre que puedas:
    – Contar con visibilidad al sistema giratorio.
    – Colocar sensores sin interferencias.
    – Modificar la distribución de masa (agregar o quitar contrapesos).

    Casos típicos donde conviene usarlo:
    La máquina rueda más de lo normal o emite sonidos extraños.
    No hay tiempo para desmontajes (operación prioritaria).
    El equipo es difícil de parar o caro de inmovilizar.
    Trabajas en campo abierto o lugares sin talleres cercanos.

    Ventajas clave vs. llamar a un técnico
    | Equipo portátil | Servicio externo |
    |—————-|——————|
    | ✔ Sin esperas (acción inmediata) | ❌ Demoras por agenda y logística |
    | ✔ Monitoreo preventivo (evitas fallas mayores) | ❌ Suele usarse solo cuando hay emergencias |
    | ✔ Reducción de costos operativos con uso continuo | ❌ Costos recurrentes por servicios |

    ¿Qué máquinas se pueden equilibrar?
    Cualquier sistema rotativo, como:
    – Turbinas de vapor/gas
    – Motores industriales
    – Ventiladores de alta potencia
    – Molinos y trituradoras
    – Hélices navales
    – Bombas centrífugas

    Requisito clave: acceso suficiente para medir y corregir el balance.

    Tecnología que simplifica el proceso
    Los equipos modernos incluyen:
    Aplicaciones didácticas (para usuarios nuevos o técnicos en formación).
    Diagnóstico instantáneo (visualización precisa de datos).
    Autonomía prolongada (ideales para trabajo en campo).

    Ejemplo práctico:
    Un molino en una mina mostró movimientos inusuales. Con un equipo portátil, el técnico identificó el problema en menos de media hora. Lo corrigió añadiendo contrapesos y impidió una interrupción prolongada.

    ¿Por qué esta versión es más efectiva?
    – Estructura más dinámica: Organización visual facilita la comprensión.
    – Enfoque práctico: Ofrece aplicaciones tangibles del método.
    – Lenguaje persuasivo: Frases como “kit de supervivencia” o “minimizas riesgos importantes” refuerzan el valor del servicio.
    – Detalles técnicos útiles: Se especifican requisitos y tecnologías modernas.

    ¿Necesitas ajustar el tono (más comercial) o añadir keywords específicas? ¡Aquí estoy para ayudarte! ️

    Reply

  994. Comercializamos equipos de equilibrio!
    Fabricamos directamente, elaborando en tres naciones simultáneamente: Argentina, España y Portugal.
    ✨Nuestros equipos son de muy alta calidad y debido a que somos productores directos, nuestro precio es inferior al de nuestros competidores.
    Hacemos entregas internacionales sin importar la ubicación, consulte los detalles técnicos en nuestra plataforma digital.
    El equipo de equilibrio es portátil, de bajo peso, lo que le permite ajustar cualquier elemento giratorio en diversos entornos laborales.

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