Calculus: Early Transcendentals, Metric Edition

James Stewart, Daniel K. Clegg, Saleem Watson

Chapter 7

Techniques of Integration - all with Video Answers

Educators

+ 4 more educators

Section 1

Integration by Parts

Problem 1

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$.
$$
\int x e^{2 x} d x ; \quad u=x, d v=e^{2 x} d x
$$

Khushbu Rani

Khushbu Rani

Numerade Educator

Problem 2

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$.
$$
\int \sqrt{x} \ln x d x ; \quad u=\ln x, d v=\sqrt{x} d x
$$

Gregory Higby

Numerade Educator

Problem 3

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$.
$$
\int x \cos 4 x d x ; \quad u=x, d v=\cos 4 x d x
$$

Linda Hand

Numerade Educator

Problem 4

Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v$.
$$
\int \sin ^{-1} x d x ; \quad u=\sin ^{-1} x, d v=d x
$$

Linda Hand

Numerade Educator

Problem 5

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

Linda Hand

Numerade Educator

Problem 6

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

Linda Hand

Numerade Educator

Problem 7

Evaluate the integral.
$$
\int x \sin 10 x d x
$$

Linda Hand

Numerade Educator

Problem 8

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

Linda Hand

Numerade Educator

Problem 9

Evaluate the integral.
$$
\int w \ln w d w
$$

Willis James

Numerade Educator

Problem 10

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

Linda Hand

Numerade Educator

Problem 11

Evaluate the integral.
$$
\int\left(x^{2}+2 x\right) \cos x d x
$$

Nafis Fuad

Numerade Educator

Problem 12

Evaluate the integral.
$$
\int t^{2} \sin \beta t d t
$$

Robert Daugherty

Robert Daugherty

Numerade Educator

Problem 13

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

Nafis Fuad

Numerade Educator

Problem 14

Evaluate the integral.
$$
\int \ln \sqrt{x} d x
$$

Lucas Gagne

Numerade Educator

Problem 15

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

Nafis Fuad

Numerade Educator

Problem 16

Evaluate the integral.
$$
\int \tan ^{-1}(2 y) d y
$$

Nafis Fuad

Numerade Educator

Problem 17

Evaluate the integral.
$$
\int t \csc ^{2} t d t
$$

Nafis Fuad

Numerade Educator

Problem 18

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

Linda Hand

Numerade Educator

Problem 19

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

Amy Jiang

Numerade Educator

Problem 20

Evaluate the integral.
$$
\int \frac{z}{10^{z}} d z
$$

Nafis Fuad

Numerade Educator

Problem 21

Evaluate the integral.
$$
\int e^{3 x} \cos x d x
$$

Linda Hand

Numerade Educator

Problem 22

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

Linda Hand

Numerade Educator

Problem 23

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

Steven Clarke

Numerade Educator

Problem 24

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

Fuzail Shakir

Numerade Educator

Problem 25

Evaluate the integral.
$$
\int z^{3} e^{z} d z
$$

Nafis Fuad

Numerade Educator

Problem 26

Evaluate the integral.
$$
\int(\arcsin x)^{2} d x
$$

Linda Hand

Numerade Educator

Problem 27

Evaluate the integral.
$$
\int\left(1+x^{2}\right) e^{3 x} d x
$$

Linda Hand

Numerade Educator

Problem 28

Evaluate the integral.
$$
\int_{0}^{1 / 2} \theta \sin 3 \pi \theta d \theta
$$

Linda Hand

Numerade Educator

Problem 29

Evaluate the integral.
$$
\int_{0}^{1} x 3^{x} d x
$$

Melinda Mulcahy

Melinda Mulcahy

Numerade Educator

Problem 30

Evaluate the integral.
$$
\int_{0}^{1} \frac{x e^{x}}{(1+x)^{2}} d x
$$

Amelia Hardy

Numerade Educator

Problem 31

Evaluate the integral.
$$
\int_{0}^{2} y \sinh y d y
$$

Linda Hand

Numerade Educator

Problem 32

Evaluate the integral.
$$
\int_{1}^{2} w^{2} \ln w d w
$$

Nafis Fuad

Numerade Educator

Problem 33

Evaluate the integral.
$$
\int_{1}^{5} \frac{\ln R}{R^{2}} d R
$$

Nafis Fuad

Numerade Educator

Problem 34

Evaluate the integral.
$$
\int_{0}^{2 \pi} t^{2} \sin 2 t d t
$$

Nafis Fuad

Numerade Educator

Problem 35

Evaluate the integral.
$$
\int_{0}^{\pi} x \sin x \cos x d x
$$

Nafis Fuad

Numerade Educator

Problem 36

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

Fuzail Shakir

Numerade Educator

Problem 37

Evaluate the integral.
$$
\int_{1}^{5} \frac{M}{e^{M}} d M
$$

Nafis Fuad

Numerade Educator

Problem 38

Evaluate the integral.
$$
\int_{1}^{2} \frac{(\ln x)^{2}}{x^{3}} d x
$$

Laurel Weber

Numerade Educator

Problem 39

Evaluate the integral.
$$
\int_{0}^{\pi / 3} \sin x \ln (\cos x) d x
$$

Nafis Fuad

Numerade Educator

Problem 40

Evaluate the integral.
$$
\int_{0}^{1} \frac{r^{3}}{\sqrt{4+r^{2}}} d r
$$

Robert Daugherty

Robert Daugherty

Numerade Educator

Problem 41

Evaluate the integral.
$$
\int_{0}^{\pi} \cos x \sinh x d x
$$

Linda Hand

Numerade Educator

Problem 42

Evaluate the integral.
$$
\int_{0}^{t} e^{s} \sin (t-s) d s
$$

James Kiss

Numerade Educator

Problem 43

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

Linda Hand

Numerade Educator

Problem 44

First make a substitution and then use integration by parts to evaluate the integral.
$$
\int \cos (\ln x) d x
$$

Linda Hand

Numerade Educator

Problem 45

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 46

First make a substitution and then use integration by parts to evaluate the integral.
$$
\int_{0}^{\pi} e^{\cos t} \sin 2 t d t
$$

Willis James

Numerade Educator

Problem 47

First make a substitution and then use integration by parts to evaluate the integral.
$$
\int x \ln (1+x) d x
$$

Lucas Gagne

Numerade Educator

Problem 48

First make a substitution and then use integration by parts to evaluate the integral.
$$
\int \frac{\arcsin (\ln x)}{x} d x
$$

Nafis Fuad

Numerade Educator

Problem 49

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

Amelia Hardy

Numerade Educator

Problem 50

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 51

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative $($ take $C=0)$.
$$
\int x^{3} \sqrt{1+x^{2}} d x
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 52

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

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 53

(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 $\int \sin ^{4} x d x$

Robert Daugherty

Robert Daugherty

Numerade Educator

Problem 54

(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 $\int \cos ^{4} x d x$.

Robert Daugherty

Robert Daugherty

Numerade Educator

Problem 55

(a) Use the reduction formula in Example 6 to show that
$$\int_{0}^{\pi / 2} \sin ^{n} x d x=\frac{n-1}{n} \int_{0}^{\pi / 2} \sin ^{n-2} x d x$$
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,
$$
\int_{0}^{\pi / 2} \sin ^{2 n+1} x d x=\frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot 2 n}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot(2 n+1)}
$$

JO

Jorge Olivares

Numerade Educator

Problem 56

Prove that, for even powers of sine,
$$
\int_{0}^{\pi / 2} \sin ^{2 n} x d x=\frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot(2 n-1)}{2 \cdot 4 \cdot 6 \cdots \cdots 2 n} \frac{\pi}{2}
$$

Robert Daugherty

Robert Daugherty

Numerade Educator

Problem 57

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

Lucas Gagne

Numerade Educator

Problem 58

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

Robert Daugherty

Robert Daugherty

Numerade Educator

Problem 59

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

Robert Daugherty

Robert Daugherty

Numerade Educator

Problem 60

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

Robert Daugherty

Robert Daugherty

Numerade Educator

Problem 61

Use Exercise 57 to find $\int(\ln x)^{3} d x$.

Steven Clarke

Numerade Educator

Problem 62

Use Exercise 58 to find $\int x^{4} e^{x} d x$.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 63

Find the area of the region bounded by the given curves.
$$
y=x^{2} \ln x, \quad y=4 \ln x
$$

WZ

Wen Zheng

Numerade Educator

Problem 64

Find the area of the region bounded by the given curves.
$$
y=x^{2} e^{-x}, \quad y=x e^{-x}
$$

Linda Hand

Numerade Educator

Problem 65

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.
$$
y=\arcsin \left(\frac{1}{2} x\right), \quad y=2-x^{2}
$$

WZ

Wen Zheng

Numerade Educator

Problem 66

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.
$$
y=x \ln (x+1), \quad y=3 x-x^{2}
$$

WZ

Wen Zheng

Numerade Educator

Problem 67

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

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 67

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

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 67

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

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 68

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

WZ

Wen Zheng

Numerade Educator

Problem 69

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

WZ

Wen Zheng

Numerade Educator

Problem 70

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.
$$
y=e^{x}, x=0, y=3 ; \quad \text { about the } x \text { -axis }
$$

WZ

Wen Zheng

Numerade Educator

Problem 71

Calculate the volume generated by rotating the region bounded by the curves $y=\ln x, y=0$, and $x=2$ about each axis.
(a) The $y$ -axis
(b) The $x$ -axis

WZ

Wen Zheng

Numerade Educator

Problem 72

Calculate the average value of $f(x)=x \sec ^{2} x$ on the interval $[0, \pi / 4]$.

WZ

Wen Zheng

Numerade Educator

Problem 73

The Fresnel function $S(x)=\int_{0}^{x} \sin \left(\frac{1}{2} \pi t^{2}\right) d t$ was discussed in Example $5.3 .3$ and is used extensively in the theory of optics. Find $\int S(x) d x$. [Your answer will involve $\left.S(x) .\right]$

James Kiss

Numerade Educator

Problem 74

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 $v_{e}$ (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 (a) one minute after liftoff and (b) after it has consumed $6000 \mathrm{~kg}$ of fuel.

Pawan Yadav

Numerade Educator

Problem 75

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 76

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 77

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 78

(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
$$\int_{a}^{b} f(x) d x=b f(b)-a f(a)-\int_{f(a)}^{f(b)} g(y) d y$$
[Hint: Use part (a) and make the substitution $y=f(x)$.]
(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 $\int_{1}^{e} \ln x d x$.

Noah Mekonnen

Numerade Educator

Problem 79

(a) Recall that the formula for integration by parts is obtained from the Product Rule. Use similar reasoning to obtain the following integration formula from the Quotient Rule.
$$\int \frac{u}{v^{2}} d v=-\frac{u}{v}+\int \frac{1}{v} d u$$
(b) Use the formula in part (a) to evaluate $\int \frac{\ln x}{x^{2}} d x$.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 80

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 56 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 55 and 56 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:
$$\frac{\pi}{2}=\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \ldots$$
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.

James Kiss

Numerade Educator

Problem 81

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
$$
V=\int_{a}^{b} 2 \pi x f(x) d x
$$

SB

Spencer Bahr

Numerade Educator