Student Solutions Manual for Stewart's / Single Variable Calculus: Early Transcendentals

James Stewart

Chapter 7

Techniques of Integration - all with Video Answers

Educators

Section 1

Integration by Parts

Problem 1

1-2 Indicated choices of $u$ and $d v$.
$\int x \ln x d x ; \quad u=\ln x, d v=x d x$

Foster Wisusik

Foster Wisusik

Numerade Educator

Problem 2

Indicated choices of $u$ and $d v$.
$\int \theta \sec ^2 \theta d \theta ; \quad u=\theta, d v=\sec ^2 \theta d \theta$

Foster Wisusik

Numerade Educator

Problem 3

3-32 Evaluate the integral.
$\int x \cos 5 x d x$

Lucas Gagne

Numerade Educator

Problem 4

Evaluate the integral.
$\int x e^{-x} d x$

Amy Jiang

Numerade Educator

Problem 5

Evaluate the integral.
$\int r e^{r / 2} d r$

Vikash Ranjan

Numerade Educator

Problem 6

Evaluate the integral.
$\int t \sin 2 t d t$

Vipender Yadav

Numerade Educator

Problem 7

Evaluate the integral.
$\int x^2 \sin \pi x d x$

Lucas Gagne

Numerade Educator

Problem 8

Evaluate the integral.
$\int x^2 \cos m x d x$

Fuzail Shakir

Numerade Educator

Problem 9

Evaluate the integral.
$\int \ln (2 x+1) d x$

Perry Roeder

Numerade Educator

Problem 10

Evaluate the integral.
$\int \sin ^{-1} x d x$

Willis James

Numerade Educator

Problem 11

Evaluate the integral.
$\int \arctan 4 t d t$

Willis James

Numerade Educator

Problem 12

Evaluate the integral.
$\int p^5 \ln p d p$

Fuzail Shakir

Numerade Educator

Problem 13

Evaluate the integral.
$\int(\ln x)^2 d x$

Daniel Mcconnell

Daniel Mcconnell

Numerade Educator

Problem 14

Evaluate the integral.
$\int t^3 e^t d t$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 15

Evaluate the integral.
$\int e^{2 \theta} \sin 3 \theta d \theta$

Linda Hand

Numerade Educator

Problem 16

Evaluate the integral.
$\int e^{-\theta} \cos 2 \theta d \theta$

Ashley Boni

Numerade Educator

Problem 17

Evaluate the integral.
$\int y \sinh y d y$

Willis James

Numerade Educator

Problem 18

Evaluate the integral.
$\int y \cosh a y d y$

Nick Johnson

Numerade Educator

Problem 19

Evaluate the integral.
$\int_0^\pi t \sin 3 t d t$

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 20

Evaluate the integral.
$\int_0^1\left(x^2+1\right) e^{-x} d x$

WZ

Wen Zheng

Numerade Educator

Problem 21

Evaluate the integral.
$\int_1^2 \frac{\ln x}{x^2} d x$

Linh Vu

Numerade Educator

Problem 22

Evaluate the integral.
$\int_1^4 \sqrt{t} \ln t d t$

JH

J Hardin

Numerade Educator

Problem 23

Evaluate the integral.
$\int_0^1 \frac{y}{e^{2 y}} d y$

Linh Vu

Numerade Educator

Problem 24

Evaluate the integral.
$\int_{\pi / 4}^{\pi / 2} x \csc ^2 x d x$

Steven Clarke

Numerade Educator

Problem 25

Evaluate the integral.
$\int_0^{1 / 2} \cos ^{-1} x d x$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 26

Evaluate the integral.
$\int_0^1 x 5^x d x$

Linda Hand

Numerade Educator

Problem 27

Evaluate the integral.
$\int \cos x \ln (\sin x) d x$

Steven Clarke

Numerade Educator

Problem 28

Evaluate the integral.
$\int_1^{\sqrt{3}} \arctan (1 / x) d x$

Fuzail Shakir

Numerade Educator

Problem 29

Evaluate the integral.
$\int \cos (\ln x) d x$

Perry Roeder

Numerade Educator

Problem 30

Evaluate the integral.
$\int_0^1 \frac{r^3}{\sqrt{4+r^2}} d r$

WZ

Wen Zheng

Numerade Educator

Problem 31

Evaluate the integral.
$\int_1^2 x^4(\ln x)^2 d x$

WZ

Wen Zheng

Numerade Educator

Problem 32

Evaluate the integral.
$\int_0^t e^x \sin (t-s) d s$

Nafis Fuad

Numerade Educator

Problem 33

33-36 First make a substitution and then use integration by parts to evaluate the integral.
$\int \sin \sqrt{x} d x$

Uma Kumari

Numerade Educator

Problem 34

First make a substitution and then use integration by parts to evaluate the integral.
$\int_1^4 e^{\sqrt{x}} d x$

WZ

Wen Zheng

Numerade Educator

Problem 35

First make a substitution and then use integration by parts to evaluate the integral.
$\int_{\sqrt{\pi / 2}}^{\sqrt{\pi}} \theta^3 \cos \left(\theta^2\right) d \theta$

Lucas Gagne

Numerade Educator

Problem 36

First make a substitution and then use integration by parts to evaluate the integral.
$\int x^5 e^{x^2} d x$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 37

37-40 Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take $C=0$ ).
$\int x \cos \pi x d x$

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 38

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take $C=0$ ).
$\int x^{3 / 2} \ln x d x$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 39

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take $C=0$ ).
$\int(2 x+3) e^x d x$

Linda Hand

Numerade Educator

Problem 40

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take $C=0$ ).
$\int x^3 e^{x^2} d x$

Linda Hand

Numerade Educator

Problem 41

(a) Use the reduction formula in Example 6 to show that
$$
\int \sin ^2 x d x=\frac{x}{2}-\frac{\sin 2 x}{4}+C
$$
(b) Use part (a) and the reduction formula to evaluate $\int \sin ^4 x d x$

Willis James

Numerade Educator

Problem 42

(a) Prove the reduction formula
$$
\int \cos ^n x d x=\frac{1}{n} \cos ^{n-1} x \sin x+\frac{n-1}{n} \int \cos ^{n-2} x d x
$$
(b) Use part (a) to evaluate $\int \cos ^2 x d x$.
(c) Use parts (a) and (b) to evaluate $\int \cos ^4 x d x$.

Fuzail Shakir

Numerade Educator

Problem 43

(a) Use the reduction formula in Example 6 to show that
$$
\int_0^{\pi / 2} \sin ^2 x d x=\frac{n-1}{n} \int_0^{\pi / 2} \sin ^{n-2} x d x
$$
where $n \geqslant 2$ is an integer.
(b) Use part (a) to evaluate $\int_0^{\pi / 2} \sin ^3 x d x$ and $\int_0^{\pi / 2} \sin ^5 x d x$.
(c) Use part (a) to show that, for odd powers of sine,
$$
\int_0^{\pi / 2} \sin ^{2 n+1} x d x=\frac{2 \cdot 4 \cdot 6 \cdots \cdot 2 n}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot(2 n+1)}
$$

Willis James

Numerade Educator

Problem 44

Prove that, for even powers of sine,
$$
\int_0^{\pi / 2} \sin ^{2 n} x d x=\frac{1 \cdot 3 \cdot 5 \cdots \cdots(2 n-1)}{2 \cdot 4 \cdot 6 \cdots \cdot 2 n} \frac{\pi}{2}
$$

Mengchun Cai

Numerade Educator

Problem 45

45-48 Use integration by parts to prove the reduction formula.
$\int(\ln x)^n d x=x(\ln x)^n-n \int(\ln x)^{n-1} d x$

Linda Hand

Numerade Educator

Problem 46

Use integration by parts to prove the reduction formula.
$\int x^n e^x d x=x^n e^x-n \int x^{n-1} e^x d x$

Fuzail Shakir

Numerade Educator

Problem 47

Use integration by parts to prove the reduction formula.
$\int\left(x^2+a^2\right)^x d x$
$$
=\frac{x\left(x^2+a^2\right)^n}{2 n+1}+\frac{2 n a^2}{2 n+1} \int\left(x^2+a^2\right)^{n-1} d x \quad\left(n \neq-\frac{1}{2}\right)
$$

Vikash Ranjan

Numerade Educator

Problem 48

Use integration by parts to prove the reduction formula.
$\int \sec ^{\approx} x d x=\frac{\tan x \sec ^{n-2} x}{n-1}+\frac{n-2}{n-1} \int \sec ^{n-2} x d x \quad(n \neq 1)$ $=0$ a $=$ a 0.

Robert Daugherty

Robert Daugherty

Numerade Educator

Problem 49

Use Exercise 45 to find $\int(\ln x)^3 d x$.

Linda Hand

Numerade Educator

Problem 50

Use Exercise 46 to find $\int x^4 e^x d x$.

Linda Hand

Numerade Educator

Problem 51

51-52 Find the area of the region bounded by the given curves.
$y=x e^{-04 x}, \quad y=0, \quad x=5$

Kian Manafi

Numerade Educator

Problem 52

Find the area of the region bounded by the given curves.
$y=5 \ln x, y=x \ln x$

Yiming Zhang

Numerade Educator

Problem 53

53-54 Use a graph to find approximate $x$-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
$y=x \sin x, y=(x-2)^2$

Adalynn Griesser

Adalynn Griesser

Numerade Educator

Problem 54

Use a graph to find approximate $x$-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
$y=\arctan 3 x, \quad y=x / 2$

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 55

55-58 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
$y=\cos (\pi x / 2), y=0,0 \leqslant x \leqslant 1$; about the $y$-axis

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 56

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
$y=e^x, y=e^{-x}, x=1$; about the $y$-axis

WZ

Wen Zheng

Numerade Educator

Problem 57

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
$y=e^{-x}, y=0, x=-1, x=0 ; \quad$ about $x=1$

WZ

Wen Zheng

Numerade Educator

Problem 58

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
$y=e^x, x=0, y=\pi$; about the $x$-axis

WZ

Wen Zheng

Numerade Educator

Problem 59

Find the average value of $f(x)=x^2 \ln x$ on the interval $[1,3]$.

Linda Hand

Numerade Educator

Problem 60

A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is $m$, the fuel is consumed at rate $r$, and the exhaust gases are ejected with constant velocity $v_{\varepsilon}$ (relative to the rocket). A model for the velocity of the rocket at time $t$ is given by the equation
$$
v(t)=-g t-v_e \ln \frac{m-r t}{m}
$$
where $g$ is the acceleration due to gravity and $t$ is not too large. If $g=9.8 \mathrm{~m} / \mathrm{s}^2, m=30,000 \mathrm{~kg}, r=160 \mathrm{~kg} / \mathrm{s}$, and $v_e=3000 \mathrm{~m} / \mathrm{s}$, find the height of the rocket one minute after liftoff.

Carson Merrill

Numerade Educator

Problem 61

A particle that moves along a straight line has velocity $v(t)=t^2 e^{-t}$ meters per second after $t$ seconds. How far will it travel during the first $t$ seconds?

WZ

Wen Zheng

Numerade Educator

Problem 62

If $f(0)=g(0)=0$ and $f^{\prime \prime}$ and $g^{\prime \prime}$ are continuous, show that
$$
\int_0^a f(x) g^{\prime \prime}(x) d x=f(a) g^{\prime}(a)-f^{\prime}(a) g(a)+\int_0^a f^{\prime \prime}(x) g(x) d x
$$

Fuzail Shakir

Numerade Educator

Problem 63

Suppose that $f(1)=2, f(4)=7, f^{\prime}(1)=5, f^{\prime}(4)=3$, and $f^{\prime \prime}$ is continuous. Find the value of $\int_1^4 x f^{\prime \prime}(x) d x$.

Lucas Gagne

Numerade Educator

Problem 64

(a) Use integration by parts to show that
$$
\int f(x) d x=x f(x)-\int x f^{\prime}(x) d x
$$
(b) If $f$ and $g$ are inverse functions and $f^{\prime}$ is continuous, prove that
$$
\int_a^b f(x) d x=b f(b)-a f(a)-\int_{f(a)}^{r(b)} g(y) d y
$$
(c) In the case where $f$ and $g$ are positive functions and $b>a>0$, draw a diagram to give a geometric interpretation of part (b).
(d) Use part (b) to evaluate $\int_1^c \ln x d x$.

Noah Mekonnen

Numerade Educator

Problem 65

We arrived at Formula 6.3.2, $V=\int_a^b 2 \pi x f(x) d x$, by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2 , at least for the case where $f$ is one-to-one and therefore has an inverse function $g$. Use the figure to show that
$$
V=\pi b^2 d-\pi a^2 c-\int_c^d \pi[g(y)]^2 d y
$$
Make the substitution $y=f(x)$ and then use integration by parts on the resulting integral to prove that $V=\int_a^b 2 \pi x f(x) d x$.
(FIGURE CAN'T COPY)

WZ

Wen Zheng

Numerade Educator

Problem 66

Let $I_n=\int_0^{\pi / 2} \sin ^n x d x$.
(a) Show that $I_{2 n+2} \leqslant I_{2 n+1} \leqslant I_{2 n}$.
(b) Use Exercise 44 to show that
$$
\frac{I_{2 n+2}}{I_{2 n}}=\frac{2 n+1}{2 n+2}
$$
(c) Use parts (a) and (b) to show that
$$
\frac{2 n+1}{2 n+2} \leqslant \frac{I_{2 n+1}}{I_{2 n}} \leqslant 1
$$
and deduce that $\lim _{n \rightarrow \infty} I_{2 n+1} / I_{2 n}=1$.
(d) Use part (c) and Exercises 43 and 44 to show that
$$
\lim _{n \rightarrow \infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \cdots \cdot \frac{2 n}{2 n-1} \cdot \frac{2 n}{2 n+1}=\frac{\pi}{2}
$$
This formula is usually written as an infinite product:
$$
\frac{\pi}{2}=\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \cdots
$$
and is called the Wallis product.
(e) We construct rectangles as follows. Start with a square of area 1 and attach rectangles of area 1 alternately beside or on top of the previous rectangle (see the figure). Find the limit of the ratios of width to height of these rectangles.
(FIGURE CAN'T COPY)

James Kiss

Numerade Educator