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Chapter 7

Techniques of Integration

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Problem 1

$1-2$ Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v .$
$$\int x^{2} \ln x d x ; \quad u=\ln x, d v=x^{2} d x$$

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Problem 2

$1-2$ Evaluate the integral using integration by parts with the indicated choices of $u$ and $d v .$
$$\int \theta \cos \theta d \theta ; \quad u=\theta, d v=\cos \theta d \theta$$

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Problem 3

$3-36$ Evaluate the integral.
$$\int x \cos 5 x d x$$

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Problem 4

$3-36$ Evaluate the integral.
$$\int y e^{0.2 y} d y$$

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Problem 5

$3-36$ Evaluate the integral.
$$\int t e^{-3 t} d t$$

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Problem 6

$3-36$ Evaluate the integral.
$$\int(x-1) \sin \pi x d x$$

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Problem 7

$3-36$ Evaluate the integral.
$$\int\left(x^{2}+2 x\right) \cos x d x$$

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Problem 8

$3-36$ Evaluate the integral.
$$\int t^{2} \sin \beta t d t$$

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Problem 9

$3-36$ Evaluate the integral.
$$\int \ln \sqrt[3]{x} d x$$

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Problem 10

$3-36$ Evaluate the integral.
$$\int \sin ^{-1} x d x$$

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Problem 11

$3-36$ Evaluate the integral.
$$\int \arctan 4 t d t$$

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Problem 12

$3-36$ Evaluate the integral.
$$\int p^{5} \ln p d p$$

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Problem 13

$3-36$ Evaluate the integral.
$$\int t \sec ^{2} 2 t d t$$

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Problem 14

$3-36$ Evaluate the integral.
$$\int s 2^{s} d s$$

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Problem 15

$3-36$ Evaluate the integral.
$$\int(\ln x)^{2} d x$$

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Problem 16

$3-36$ Evaluate the integral.
$$\int t \sinh m t d t$$

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Problem 17

$3-36$ Evaluate the integral.
$$\int e^{2 \theta} \sin 3 \theta d \theta$$

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Problem 18

$3-36$ Evaluate the integral.
$$\int e^{-\theta} \cos 2 \theta d \theta$$

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Problem 19

$3-36$ Evaluate the integral.
$$\int z^{3} e^{z} d z$$

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Problem 20

$3-36$ Evaluate the integral.
$$\int x \tan ^{2} x d x$$

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Problem 21

$3-36$ Evaluate the integral.
$$\int \frac{x e^{2 x}}{(1+2 x)^{2}} d x$$

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Problem 22

$3-36$ Evaluate the integral.
$$\int(\arcsin x)^{2} d x$$

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Problem 23

$3-36$ Evaluate the integral.
$$\int_{0}^{1 / 2} x \cos \pi x d x$$

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Problem 24

$3-36$ Evaluate the integral.
$$\int_{0}^{1}\left(x^{2}+1\right) e^{-x} d x$$

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Problem 25

$3-36$ Evaluate the integral.
$$\int_{0}^{1} t \cosh t d t$$

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Problem 26

$3-36$ Evaluate the integral.
$$\int_{4}^{9} \frac{\ln y}{\sqrt{y}} d y$$

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Problem 27

$3-36$ Evaluate the integral.
$$\int_{1}^{3} r^{3} \ln r d r$$

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Problem 28

$3-36$ Evaluate the integral.
$$\int_{0}^{2 \pi} t^{2} \sin 2 t d t$$

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Problem 29

$3-36$ Evaluate the integral.
$$\int_{0}^{1} \frac{y}{e^{2 y}} d y$$

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Problem 30

$3-36$ Evaluate the integral.
$$\int_{1}^{\sqrt{3}} \arctan (1 / x) d x$$

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Problem 31

$3-36$ Evaluate the integral.
$$\int_{0}^{1 / 2} \cos ^{-1} x d x$$

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Problem 32

$3-36$ Evaluate the integral.
$$\int_{1}^{2} \frac{(\ln x)^{2}}{x^{3}} d x$$

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Problem 33

$3-36$ Evaluate the integral.
$$\int \cos x \ln (\sin x) d x$$

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Problem 34

$3-36$ Evaluate the integral.
$$\int_{0}^{1} \frac{r^{3}}{\sqrt{4+r^{2}}} d r$$

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Problem 35

$3-36$ Evaluate the integral.
$$\int_{1}^{2} x^{4}(\ln x)^{2} d x$$

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Problem 36

$3-36$ Evaluate the integral.
$$\int_{0}^{t} e^{s} \sin (t-s) d s$$

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Problem 37

$37-42$ First make a substitution and then use integration by parts to evaluate the integral.
$$\int \cos \sqrt{x} d x$$

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Problem 38

$37-42$ First make a substitution and then use integration by parts to evaluate the integral.
$$\int^{3} e^{-t} d t$$

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Problem 39

$37-42$ First make a substitution and then use integration by parts to evaluate the integral.
$$\int_{\sqrt{\pi / 2}}^{\sqrt{\pi}} \theta^{3} \cos \left(\theta^{2}\right) d \theta$$

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Problem 40

$37-42$ First make a substitution and then use integration by parts to evaluate the integral.
$$\int_{0}^{\pi} e^{\cos t} \sin 2 t d t$$

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Problem 41

$37-42$ First make a substitution and then use integration by parts to evaluate the integral.
$$\int x \ln (1+x) d x$$

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Problem 42

$37-42$ First make a substitution and then use integration by parts to evaluate the integral.
$$\int \sin (\ln x) d x$$

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Problem 43

$43-46$ Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its anti derivative (take $C=0 )$ .
$$\int x e^{-2 x} d x$$

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Problem 44

$43-46$ Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its anti derivative (take $C=0 )$ .
$$\int x^{3 / 2} \ln x d x$$

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Problem 45

$43-46$ Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its anti derivative (take $C=0 )$ .
$$\int x^{3} \sqrt{1+x^{2}} d x$$

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Problem 46

$43-46$ Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its anti derivative (take $C=0 )$ .
$$\int x^{2} \sin 2 x d x$$

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

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

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Problem 49

(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,
$$\int_{0}^{\pi / 2} \sin ^{2 n+1} x d x=\frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot 2 n}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot(2 n+1)}$$

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Problem 50

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

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Problem 51

$51-54$ Use integration by parts to prove the reduction formula.
$$\int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x$$

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Problem 52

$51-54$ Use integration by parts to prove the reduction formula.
$$\int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x$$

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Problem 53

$51-54$ Use integration by parts to prove the reduction formula.
$$\int \tan ^{n} x d x=\frac{\tan ^{n-1} x}{n-1}-\int \tan ^{n-2} x d x \quad(n \neq 1)$$

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Problem 54

$51-54$ Use integration by parts to prove the reduction formula.
$$\int \sec ^{n} x d x=\frac{\tan x \sec ^{n-2} x}{n-1}+\frac{n-2}{n-1} \int \sec ^{n-2} x d x \quad(n \neq 1)$$

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Problem 55

Use Exercise 51 to find $\int(\ln x)^{3} d x$

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Problem 56

56. Use Exercise 52 to find $\int x^{4} e^{x} d x$

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Problem 57

$57-58$ Find the area of the region bounded by the given curves.
$$y=x^{2} \ln x, \quad y=4 \ln x$$

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Problem 58

$57-58$ Find the area of the region bounded by the given curves.
$$y=x^{2} e^{-x}, \quad y=x e^{-x}$$

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Problem 59

$59-60$ 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.
$$y=\arcsin \left(\frac{1}{2} x\right), \quad y=2-x^{2}$$

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Problem 60

$59-60$ 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.$ $$y=x \ln (x+1), \quad y=3 x-x^{2}$$

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Problem 61

$61-63$ Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
$y=\cos (\pi x / 2), y=0,0 \leqslant x \leqslant 1 ; \quad$ about the $y$ -axis

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Problem 62

$61-63$ Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
$y=e^{x}, y=e^{-x}, x=1 ;$ about the $y$ -axis

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Problem 63

$61-63$ Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
$y=e^{-x}, y=0, x=-1, x=0 ; \quad$ about $x=1$

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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

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Problem 65

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

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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 $v_{e}$ (relative to the rocket). A model for the velocity of the rocket at
time $t$ is given by the equation
$v(t)=-g t-v_{c} \ln \frac{m-r t}{m}$
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.

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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?

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Problem 68

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

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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 $\int_{1}^{4} x f^{\prime \prime}(x) d x$

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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
$\int_{a}^{b} f(x) d x=b f(b)-a f(a)-\int_{f(a)}^{(b)} g(y) d y$
[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 interpret
tion of part (b)
(d) Use part (b) to evaluate $\int_{1}^{e} \ln x d x$

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Problem 71

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
$V=\int_{0}^{b} 2 \pi x f(x) d x$

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Problem 72

Let $I_{n}=\int_{0}^{\pi / 2} \sin ^{n} X d x$
(a) Show that $I_{2 n+2} \leqslant I_{2 n+1} \leqslant I_{2 \pi}$
(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 $\lim _{n \rightarrow \infty} I_{2 n+1} / L_{2 n}=1$
(d) Use part $(\mathrm{c})$ and Exercises 49 and 50 to show that
$$\lim _{n \rightarrow \infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \ldots \cdot \frac{2 n}{2 n-1} \cdot \frac{2 n}{2 n+1}=\frac{\pi}{2}$$
This formula is usually written as an infinite product:
$$\frac{\pi}{2}=\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \dots$$
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.

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