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Calculus

James Stewart

Chapter 7

Techniques of Integration - all with Video Answers

Educators

Section 1

7.1 Exercises

03:51

Problem 1

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

Lucas Gagne
Lucas Gagne
Numerade Educator
00:40

Problem 2

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

Fuzail Shakir
Fuzail Shakir
Numerade Educator
01:08

Problem 3

Evaluate the integral.
$\int x \cos 5 x d x$

DQ
Danjoseph Quijada
Numerade Educator
02:52

Problem 4

Evaluate the integral.
$\int y e^{0.2 y} d y$

Mary Wakumoto
Mary Wakumoto
Numerade Educator
01:31

Problem 5

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

DQ
Danjoseph Quijada
Numerade Educator
01:34

Problem 6

Evaluate the integral.
$\int(x-1) \sin \pi x d x$

DQ
Danjoseph Quijada
Numerade Educator
04:39

Problem 7

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

Robert Daugherty
Robert Daugherty
Numerade Educator
View

Problem 8

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

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
03:15

Problem 9

Evaluate the integral.
$\int \ln \sqrt[3]{x} d x$

Lucas Gagne
Lucas Gagne
Numerade Educator
05:43

Problem 10

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

Willis James
Willis James
Numerade Educator
01:06

Problem 11

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

Tyler Moulton
Tyler Moulton
Numerade Educator
00:46

Problem 12

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

Fuzail Shakir
Fuzail Shakir
Numerade Educator
02:11

Problem 13

Evaluate the integral.
$\int t \sec ^2 2 t d t$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:21

Problem 14

Evaluate the integral.
$\int s 2^x d s$

Yiyang Wang
Yiyang Wang
Numerade Educator
04:02

Problem 15

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

Daniel Mcconnell
Daniel Mcconnell
Numerade Educator
02:10

Problem 16

Evaluate the integral.
$\int t \sinh m t d t$

Foster Wisusik
Foster Wisusik
Numerade Educator
04:04

Problem 17

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

Linda Hand
Linda Hand
Numerade Educator
06:54

Problem 18

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

Ashley Boni
Ashley Boni
Numerade Educator
04:10

Problem 19

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

WZ
Wen Zheng
Numerade Educator

Problem 20

Evaluate the integral.
$\int x \tan ^2 x d x$

Check back soon!
00:34

Problem 21

Evaluate the integral.

Dwijendra Rao
Dwijendra Rao
Numerade Educator
05:20

Problem 22

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

WZ
Wen Zheng
Numerade Educator
02:43

Problem 23

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

WZ
Wen Zheng
Numerade Educator
05:25

Problem 24

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

WZ
Wen Zheng
Numerade Educator
01:07

Problem 25

Evaluate the integral.
$\int_0^1 t \cosh t d t$

Adrian Co
Adrian Co
Numerade Educator
01:20

Problem 26

Evaluate the integral.
$\int_4 \frac{\ln y}{\sqrt{y}} d y$

Fuzail Shakir
Fuzail Shakir
Numerade Educator
04:08

Problem 27

Evaluate the integral.
$\int_1^3 r^3 \ln r d r$

Robert Daugherty
Robert Daugherty
Numerade Educator
05:46

Problem 28

Evaluate the integral.
$\int_0^{2 \pi} t^2 \sin 2 t d t$

Nafis Fuad
Nafis Fuad
Numerade Educator
02:01

Problem 29

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

Linh Vu
Linh Vu
Numerade Educator
04:48

Problem 30

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

WZ
Wen Zheng
Numerade Educator
02:34

Problem 31

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

Linda Hand
Linda Hand
Numerade Educator
05:23

Problem 32

Evaluate the integral.
$\int_1^2 \frac{(\ln x)^2}{x^3} d x$

WZ
Wen Zheng
Numerade Educator
02:33

Problem 33

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

Steven Clarke
Steven Clarke
Numerade Educator
03:36

Problem 34

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

WZ
Wen Zheng
Numerade Educator
06:08

Problem 35

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

WZ
Wen Zheng
Numerade Educator
03:46

Problem 36

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

Nafis Fuad
Nafis Fuad
Numerade Educator
05:18

Problem 37

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

Lucas Gagne
Lucas Gagne
Numerade Educator
01:43

Problem 38

First make a substitution and then use integration by parts to evaluate the integral.
$\int t^3 e^{-t^2} d t$

Fuzail Shakir
Fuzail Shakir
Numerade Educator
07:46

Problem 39

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
Lucas Gagne
Numerade Educator
08:18

Problem 40

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
Willis James
Numerade Educator
10:04

Problem 41

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

Lucas Gagne
Lucas Gagne
Numerade Educator
02:49

Problem 42

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

Fuzail Shakir
Fuzail Shakir
Numerade Educator
View

Problem 43

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
Amelia Hardy
Numerade Educator
01:50

Problem 44

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
01:59

Problem 45

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
02:38

Problem 46

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
09:56

Problem 47

(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
Willis James
Numerade Educator
08:10

Problem 48

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

Carson Merrill
Carson Merrill
Numerade Educator
08:52

Problem 49

. (a) Use the reduction formula in Example 6 to show that $$ \int_0^{\pi / 2} \sin ^{\varepsilon} x d x=\frac{n-1}{n} \int_0^{\pi / 2} \sin ^{n-2} x d x $$
where $n \supsetneqq 2$ 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. $$\int_0^{\pi / 2} \sin ^{2 n+1} x d x=\frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot 2 n}{3 \cdot 5 \cdot 7 \cdot \cdots\cdot(2 n+1)}$$

Willis James
Willis James
Numerade Educator
05:10

Problem 50

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

Mengchun Cai
Mengchun Cai
Numerade Educator
03:18

Problem 51

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

Lucas Gagne
Lucas Gagne
Numerade Educator
00:35

Problem 52

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
Fuzail Shakir
Numerade Educator
03:46

Problem 53

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
06:49

Problem 54

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

Robert Daugherty
Robert Daugherty
Numerade Educator
05:10

Problem 55

Use integration by parts to prove the reduction formula.
Use Exercise 51 to find $\int(\ln x)^3 d x$.

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
01:59

Problem 56

Use integration by parts to prove the reduction formula.
. Use Exercise 52 to find $\int x^4 e^x d x$.

Nafis Fuad
Nafis Fuad
Numerade Educator
01:14

Problem 57

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

Stephen Hobbs
Stephen Hobbs
Numerade Educator
05:52

Problem 58

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

WZ
Wen Zheng
Numerade Educator
05:54

Problem 59

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
05:05

Problem 60

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
04:14

Problem 61

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
03:07

Problem 62

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
04:20

Problem 63

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

WZ
Wen Zheng
Numerade Educator
03:53

Problem 64

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

Tp Sarathy
Tp Sarathy
Numerade Educator
03:46

Problem 65

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

WZ
Wen Zheng
Numerade Educator
04:02

Problem 66

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 $v_c$ (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.

Pawan Yadav
Pawan Yadav
Numerade Educator
04:28

Problem 67

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
01:48

Problem 68

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
Fuzail Shakir
Numerade Educator
03:22

Problem 69

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
Lucas Gagne
Numerade Educator
17:53

Problem 70

(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 $$\int_a^b f(x) d x=b f(b)-a f(a)-\int_{f(a)}^{f(b)} g(y) d y$$
(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
Noah Mekonnen
Numerade Educator
06:52

Problem 71

We arrived at Formula $5.3 .2, V=\int_a^b 2 \pi x f(x) d x$, by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 5.2, at least for the case where $f$ is one-to-one and therefore has an inverse function $g$. Use the figure to show that $$ V=\pi b^2 d-\pi a^2 c-\int_{\varepsilon}^d \pi[g(y)]^2 d y $$ Make the substitution $y=f(x)$ and then use integration by parts on the resulting integral to prove that $$ V=\int_a^b 2 \pi x f(x) d x $$

WZ
Wen Zheng
Numerade Educator
03:26

Problem 72

Let $I_n=\int_0^{\pi / 2} \sin ^n x d x$.
(a) Show that $I_{2 n+2} \leqslant I_{2 n+1} \leqslant I_{2 k}$.
(b) Use Exercise 50 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 $\lim _{n \rightarrow \infty} I_{2 \varepsilon+1} / I_{2 x}=1$.
(d) Use part (c) and Exercises 49 and 50 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.

James Kiss
James Kiss
Numerade Educator