Chap 38 Exercises

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Exercise 1 1.What is the differential of $u=x+5$?

$du=dx$

$du=(x+5)dx$

$du=5dx$

$du=xdx$

question id: owl-give-closet-1

2. What is the differential of $u=\sin (2x+5)$?

$du=2\cos (2x+5)dx$

$du=(2x+5)dx$

$du=5dx$

$du=2xdx$

question id: owl-give-closet-2

3. What is the differential of $v=e^x$?

$du=e^{x}dx$

$dv=e^{x}dx$

$du=xdx$

$dv=xdx$

question id: owl-give-closet-3

4. What is the differential of $f=\cos (\ln (t))$?

$df=-\frac{\sin (\ln (t))}{t}dt$

$du=-\frac{\sin (\ln (t))}{t}dt$

$dv=-\frac{\sin (\ln (t))}{t}dt$

$df=-\frac{\sin (\ln (x))}{x}dx$

question id: owl-give-closet-4

Exercise 2 Go through the steps for integration by parts in ?sec-integration-by-parts1 or ?sec-integration-by-parts2 to find the anti-derivative of $g(x)\equiv x\cos (x)$.

Step 1 hint: We know the anti-derivative of $\cos (x)$ is $\sin (x)$, so an appropriate helper function is the function $x\sin (x)$. Now do steps (2) and (3): (2) take the derivative of the helper function and then (3) integrate each term in the result.

1. What is the derivative of the helper function with respect to $x$?

${\partial }_x\text{helper}(x)=\sin (x)+x\cos (x)$

${\partial }_x\text{helper}(x)=\sin (x)+x\sin (x)$

${\partial }_x\text{helper}(x)=\sin (x)+\cos (x)$

${\partial }_x\text{helper}(x)=\sin (x)\cos (x)$

question id: bird-hold-jacket-1

2. What is $$\int {\partial }_x\text{helper}(x)?$$

$\text{helper}(x)+C$

$\frac{1}{2}{\text{helper}}^2(x)$

$1/\text{helper}(x)$

Whatever it is, it is just as complicated as the original integral. No obvious way to do it.

question id: bird-hold-jacket-2

3. What is the integral with respect to $x$ of the first part of the expanded form of the helper function, that is, $\int \sin (x)dx$?

$-\cos (x)$

$\cos (x)$

$e^x\sin (x)$

$e^x\cos (x)$

question id: bird-hold-jacket-3

4. What is the integral with respect to $x$ of the second part of the expanded form of the helper function, that is, $\int x\cos (x)$?

It is the same as the original problem! I thought you were showing us how to do the problem. If we didn’t know the answer when we started, why should we be able to do it now?

It is the same as the original problem. I’ve got an equation involving the original problem and two bits of algebra/calculus that I know how to do. Thanks!

$\sin (x)$

question id: bird-hold-jacket-4

Solve for the answer to the original function and write the function in R notation in Active R chunk 1:

Active R chunk 1

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makeFun( ...your stuff here... ~ x)

Exercise 3 Calculate each of the following anti-derivatives.

1. $\int \frac{x}{\sqrt{9-x^2}}dx=$

$\frac{1}{2}\ln \sqrt{9-x^2}+C$

$\sqrt{9-x^2}+C$

$\frac{1}{4}\sqrt{9-x^2}+C$

$\sqrt{9-x^2}+C$

question id: fir-chew-ship-1

2. $\int \frac{\ln |x|}{x}dx=$+

$+\frac{(ln|x|{)}^2}{2}+C$       $\frac{1}{2x^2}+C$       $\frac{1-\ln |x|}{x^2}+C$       $C$      

question id: fir-chew-ship-2

3. $in{t}_0^{\frac{\pi }{4}}\sin (2x)dx=$

-1.5       0.5       11      

question id: fir-chew-ship-3

4. ${\int }_0^{\frac{\pi }{2}}\cos (x)\sqrt{\sin (x)}dx=$

$-\frac{2}{3}$       $-\frac{1}{2}$       $\frac{1}{2}$       $\frac{2}{3}$      

question id: fir-chew-ship-4

5. $\int {\sin }^3(5x)\cdot \cos (5x)dx=\int (\sin (5x){)}^3\cdot \cos (5x)dx=$

$frac{\sin }^2(5x)10+C$       $frac{\sin }^4(5x)20+C$       $frac{\sin }^4(5x)4+C$       $frac4{\sin }^4(5x)5+C$      

question id: fir-chew-ship-5

Exercise 4 Compute symbolically the following anti-derivatives. 1. $(x (x) ) dx=

$ x^{}(x)- x^{}+C $

$ x^{}(x)-x{}+C $

$\frac{2}{3}{x}^{\frac{3}{2}}\cdot \ln (x)-\frac{4}{9}{x}^{\frac{3}{2}}+C$

$-2 x^{-}(x)-4 x^-{}+C $

question id: fawn-spend-futon-1

2. $-3x(2x) dx=

$-\frac{3}{2}\cos (2x)-3x\cdot \sin (2x)+C$

$-\frac{3}{4}\cos (2x)-\frac{3}{2}\sin (2x)+C$

$-\frac{3}{4}\cos (2x)-\frac{3}{2}x\cdot \sin (2x)+C$

$-\frac{3}{4}\cos (2x)-\frac{3}{2}x\cdot \sin (2x)+C$

question id: fawn-spend-futon-2

3. $\int (y^3\cdot e^y)dy$

$\frac{1}{4}{y}^4{e}^y+C$

$y^3{e}^y-3e^y{y}^2+C$

$y^3{e}^y+3y^2{e}^y-6y{e}^y+6e^y+C$

$y^3{e}^y-3y^2{e}^y+6y{e}^y-6e^y+C$

question id: fawn-spend-futon-3

Exercise 5 Compute symbolically the following anti-derivatives.

1. $\int \frac{\ln (x)}{x^3}dx=$

$\frac{4\cdot (x\ln (x)-x)}{x^4}+C$

$\frac{x^3\cdot \frac{1}{x}-ln(x)\cdot 3x^2}{x^6}+C$

$-\frac{1}{2}{x}^{-2}\cdot \ln (x)+\frac{1}{2}{x}^{-3}+C$

$-\frac{\ln (x)}{2x^2}-\frac{1}{4x^2}+C$

question id: fly-tell-drawer-1

$2.\int x^3\cdot \cos (x^4+2)dx=$

$\frac{\sin (x^4+2)}{4}+C$

$3x^2\cdot \cos (x^4+2)-4x^6\cdot \sin (x^4+2)+C$

$\sin (x^4+2)+C$

$4\sin (x^4+2)+C$

question id: fly-tell-drawer-2

3. $\int e^t\cdot (2e^t-6{)}^{3}dt=$

$e^t\cdot (2e^t-6{)}^3+3e^{2t}\cdot (2e^t-6{)}^2+C$

$(2e^t-6{)}^4+C$

$\frac{(2e^t-6{)}^4}{8}+C$

$\frac{(2e^t-6{)}^4}{4}+C$

question id: fly-tell-drawer-3

4. $\int \frac{x}{e^{3x}}dx=$

$\frac{e^{3x}-3x\cdot e^{3x}}{e^{6x}}+C$

$-\frac{x}{3}{e}^{-3x}-\frac{1}{9}{e}^{-3x}+C$

$\frac{3x^2}{2e^{3x}}+C$

$x\cdot e^{-3x}-e^{-3x}+C$

question id: fly-tell-drawer-4

5. ${\int }_0^{\pi }{\cos }^4(x)\cdot \sin (x)dx={\int }_0^{\pi }(\cos (x){)}^4\cdot \sin (x)dx=$

$-\frac{2}{5}$

$0$

$\frac{2}{5}$

$2$

question id: fly-tell-drawer-5

6. $\int (x+3)(x^2-1)dx=$

$(x^2-1)+2x\cdot (x+3)+C$

$(\frac{x^2}{2}+3x)(\frac{x^3}{3}-x)+C$

$3x^2+6x-1+C$

$\frac{x^4}{4}+x^3-\frac{x^2}{2}-3x+C$

question id: fly-tell-drawer-6

Exercise 6 Find the anti-derivative of $x\cos (2x)$. Since $$\cos (2x)=\frac{1}{2}{\partial }_x\sin (2x)$$ a reasonable guess for a helper function will be $x\sin (2x)$.

(We have intentionally dropped the $1/2$ to simplify the rest of the procedure. You will see that such multiplicative constants don’t matter, since they will be on both sides of the equation showing the derivative of the helper function. You can see this by keeping the $1/2$ in the helper function and watching what happens to it.)

As you work through the steps be very careful about the constants and make sure you check your final answer by differentiating.

1. What is ${\partial }_{x}x\sin (x)$?

$\sin (x)+x\cos (x)$

$\sin (x)+x\sin (x)$

$\cos (x)+x\sin (x)$

$\cos (x)+x\cos (x)$

question id: chicken-cost-coat-1

2. What is $\int {\partial }_x[x\sin (x)]dx$?

$x\sin (x)+C$

$\sin (x)+C$

$\cos (x)+C$

$x\cos (x)+C$

question id: chicken-cost-coat-2

3. What is $\int x\cos (x)dx$?

$x\sin (x)+\cos (x)$

$x\cos (x)+\cos (x)$

$x\cos (x)+\sin (x)$

$x\sin (x)+\sin (x)$

question id: chicken-cost-coat-3

Exercise 7 Looking for interior functions for U-substitution …

1. In $h(x)=2x/(x^2+2)$ which of the following is a plausible candidate for an interior function $g(x)$?

$\sin (x)$

$\ln (x)$

$2x$

$x^2+2$

question id: frog-pitch-clock-1

2. Continuing with the integral of $h(x)=2x/(x^2+2)$ and the working guess that $g(x)=x^2+2$, do you see any part of $h()$ which is a match to ${\partial }_{x}g()$?

$1/x$

$\ln (x)$

$2x$

question id: frog-pitch-clock-2

3. Taking seriously the progress we made in the previous two questions, we now need to write $h(x)$ as $f(x^2+2)2x$? What should $f()$ be to make this match $h(x)$?

$f(x)=\sin (x)$

$f(x)=\ln (x)$

$f(x)=1/x$

$x^2+2$

question id: frog-pitch-clock-3

Now that you have found both $g()$ and $f()$, you simply need to find a function $F(x)$ such that ${\partial }_{x}F(x)=f(x)$. Since ${\partial }_x\ln (x)=1/x$, we know that $F(x)=\ln (x)$. Thus, $\int h(x)dx=F(g(x))=F(x^2+2)=\ln (x^2+2)$.

Exercise 8 Regarding U-substitution ….

1. $\text{Find a plausible interior g(x) in}x\exp (x^2+3)$

$\exp (x)$

$x$

$x^2+3$

$x^2$

question id: lamb-mean-pan-1

2. Using your candidate for $g()$ from the previous question, which of these is a exterior f(x) in $x\exp (x^2+3)$

$f(x)=\exp (x)$

$f(x)=x$

$f(x)=x\exp (x)$

$f(x)=\ln (x)$

question id: lamb-mean-pan-2

Confirm that $h(x)=f(g(x)){\partial }_{x}g(x)$ and you win. The answer will be $F(g(x))+C$

Exercise 9 A giant tortoise (with very good eyesight and standing on an unobstructed plane!) spies a head of lettuce on the ground 65 meters away. Being hungry (and knowing the shortest path between two points on the plane!), the tortoise takes off in a straight line for the lettuce. She pretty quickly reaches her top speed, but then starts to tire. If her velocity as a function of time (in meters per minute) is modeled by $v(t)=7t{e}^{-0.3t}$, how long does it take the tortoise to reach her lunch? Answer this question by finding an calculus/algebra formula for the tortoise’s displacement and then use it to approximate how long it takes to get to the lettuce.

We will be looking at $\int v(t)dt=7\int t{e}^{-0.3t}dt$.

We will call the left side of the equation “displacement(t)”. Use integration by parts to find displacement(t) as a simple formula in $t$.

The tortoise will reach the cabbage at time $t^{\star }$ such that $\text{displacement}(t^{\star })=65$ meters. Use Active R chunk 2 to graph your displacement function to find $t^{\star }$. Remember that there will be a constant of integration which can be found by noting that $\text{displacement}(0)=0$. (Note that the graphics domain given in the example code isn’t necessarily the best choice for answering the question.)

Active R chunk 2

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2

displacement <- makeFun(7 * WHAT + C ~ t)

slice_plot(displacement(t) ~ t, bounds(t=c(0,5)))

At what time $t^{\star }$ does the tortoise reach the cabbage?

5.95 sec

10.85 sec

15.75 sec

Never! (That is, $t^{\star }$ is infinite.)

question id: crow-eat-bottle-1

Exercise 10 Which of these candidates for $f()$ and $g()$ will produce $$f(g(x)){\partial }_{x}g(x)=x^3\cos (x^4)?$$

$f(x)=\cos (x)/4$ and $g(x)=x^4$

$f(x)=\cos (x)$ and $g(x)=x^4$

$f(x)=x^4$ and $g(x)=\cos (x)$

question id: shark-dive-hamper-1

Once again, $\int h(x)dx=F(g(x))$, where ${\partial }_{x}F(x)=f(x)$.

Exercise 11 Compute the value of the definite integral $${\int }_0^{1}x\ln (x)dx.$$ First, compute the anti-derivative using integration by parts, then evaluate the anti-derivative at the lower and upper bounds of integration.

As part of your work, you will need to evaluate ${x^2\ln (x)|}_{x=0}$. This is an indeterminate form, since $x=0$ is not in the domain of $\ln (x)$. So use $\underset{x\to 0}{lim}[x^2\ln (x)]$ instead Plotting out $x^2\ln (x)$ over the domain $0<x\le {10}^{-4}$ will guide your answer.

What is the value of ${\int }_0^{1}x\ln (x)dx$?

$x^2(2\ln (x)-1)/4$

-0.25

0

0.25

question id: wolf-jump-sofa-1

Exercise 12 What is $$\int \frac{\sin (x)}{{\cos }^5(x)}dx?$$

$\ln (cos(x))$

$-\frac{1}{4}{\cos }^{-4}(x)$

$\frac{1}{6}{\cos }^{-6}(x)$

question id: No3exI

Exercise 13 Use u-substitution to find $$\int \frac{4e^{4x}+4}{e^{4x}}dx$$

$\ln (e^{4x}+4)$

$1/(e^{4x}+4)$

$\frac{1}{4}{e}^{4x}+4$

$\frac{1}{2}1/(e^{4x}+4{)}^2$

question id: cat-go-window-1

Exercise 14 Tables of integrals

Although any function has an anti-derivative, that anti-derivative cannot always be presented in algebraic notation. This poses no fundamental problem to the construction of the anti-derivative, particularly when a computer is available to handle the book-keeping of numerical integration.

Still, it is convenient to have an algebraic form when it can be found. Many people have devoted considerable effort to constructing extensive collections of functions for which an algebraic form of anti-derivative is known. Think of such collections as a gallery of portraits of people who happen to have red hair. No matter how large the collection, you will often have to deal people who are not redheads. And unlike real redheads, it can be hard to know whether a function has an anti-derivative that can be expressed simply in algebraic form. For instance, $f(x)\equiv \exp (-x^2)$ does not, even though it is ubiquitous in fields such as statistics.

So, how to organize the gallery of redheads? Let’s take a field trip to the NIST DLMF (The US National Institute of Standards and Technology (NIST) has been a primary publisher for more than 50 years of information about functions encountered in applied mathematics. The work, published originally in book form, is also available via the internet as the NIST Digital Library of Mathematical Functions!

Warning! Many visitors to NIST DLMF encounter dizziness, fatigue, and anxiety. Should you experience such symptoms, close your eyes and remember that DLMF is a reference work and that you will not be examined on its use. Nonetheless, to help you benefit maximally from the field trip, there are a few questions in this Daily Digital for you to answer from DLMF.

You should also note that the techniques in almost universal use to help you navigate through voluminous collections of data (e.g. Twitter, Facebook, Instagram, YouTube) such as ratings, subscribing, “friending,” following, etc. are entirely absent from DLMF. There is not even a friendly introduction to each chapter saying who the material might be of interest to.

We will focus on Chapter 4, “Elementary Functions,” and indeed just a few sections from that chapter. (A better name for the chapter would be “The Functions Most Often Used.” They are not “elementary” as in “elementary school” but as in the “periodic table of elements.”)

Section 4.10 covers integrals and anti-derivatives of logarithmic, exponential and power-law functions.

Section 4.26 is similar, but for trigonometric functions.

1. Navigate to equation 4.10.1. This is one of the anti-derivatives you are expected to know by heart in CalcZ.
   1. Notice that the input name $z$ is used. They could have selected any other name; $x$ and $t$ are popular, $y$ less so, and $\xi$ even less so. The use of $z$ is a signal to the cognoscenti that the function can be applied to both real and complex numbers.
   2. Look at equation 4.10.8. This is another of the functions whose anti-derivative you should know by heart.
   3. Perhaps it would have avoided some confusion if 4.10.1 had been written in terms of $$\int \frac{1}{az}dz$$ so that you would know what to do if you had encountered such a function.

Some exercises:

1. Which of these is $\int \frac{1}{az}dz$?

$\frac{1}{a}\ln (z)$

There is no anti-derivative of $1/az$.

$a\ln (z)$

question id: wolf-come-room-1

2. Use Section 4.10 of DLMF to find $\int \frac{1}{e^{3t}+5}dt$

$\frac{1}{15}(3t-\ln (e^{3t}+5))$

There is no such function listed in Section 4.10.

$\frac{1}{15}(5t-\ln (e^{5t}+3))$

question id: wolf-come-room-2

3. Is $\frac{1}{e^{az}+b}$ different from $e^{-(az+b)}$?

Yes

No

Depends on the value of $b$.

question id: wolf-come-room-3

4. Using section 4.26, find $\int \tan (\theta )d\theta$.

$-\ln (\cos (\theta ))$

$\tan (\theta )$ does not have an anti-derivative.

There is no $\theta$ in section 4.26

question id: wolf-come-room-4

Activities

Exercise 15 Using integration by parts, show that $$\int x^2{e}^{-x}dx=e^{-x}(2-2x-x^2)$$ Hint: The 2’s in $2-2x-x^2$ are a hint that you don’t need to integrate $x^2$.

Show your work.

Exercise 16 The integral $$\int x\sin (x)dx=-x\cos (x)+\sin (x)+C.$$

Confirm this result by differentiating both side of the above equation.

As well, use the given result to figure out what was the choice of $u(x)$ and $v^{\prime }(x)dx$ when $\int x\sin (x)dx$ was integrated by parts.

Show your work.

Exercise 17 Use integration by parts to find $$\int x^2\ln (x)dx.$$ Show your choices of $u(x)$ and $v^{\prime }(x)dx$.

Exercise 18 Use integration by parts to find $\int \ln (x)dx$. Show your work.

38-exercises.rmarkdown

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