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Chapter 6

TECHNIQUES OF INTEGRATION

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Problem 1

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

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Problem 2

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

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Problem 3

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

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Problem 4

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

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Problem 5

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

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Problem 6

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

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Problem 7

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

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Problem 8

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

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Problem 9

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

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Problem 10

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

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Problem 11

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

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Problem 12

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

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Problem 13

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

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Problem 14

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

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Problem 15

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

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Problem 16

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

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Problem 17

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

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Problem 18

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

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Problem 19

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

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Problem 20

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

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Problem 21

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

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Problem 22

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

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Problem 23

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

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Problem 24

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

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Problem 25

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

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Problem 26

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

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Problem 27

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 28

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

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Problem 29

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 30

First make a substitution and then use integration by
parts to evaluate the integral.
$$
\int_{1}^{4} e^{\sqrt{x}} d x
$$

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Problem 31

(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 32

(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 33

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

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Problem 34

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 35

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 36

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 37

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 38

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

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Problem 39

Use Exercise 35 to find $\int(\ln x)^{\prime} d x$

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Problem 40

Use Exercise 36 to find $\int x^{4} e^{x} d x$

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Problem 41

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

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Problem 42

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 exhast 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
$v_{c}=3000 \mathrm{m} / \mathrm{s}$ , find the height of the rocket one minute
after liftoff.

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Problem 43

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

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Problem 44

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 45

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 46

(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
$$
\int_{a}^{b} f(x) d x=b f(b)-a f(a)-\int_{f(a)}^{(j)} g(y) d y
$$
[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 cvaluate $\int_{1}^{c} \ln x d x$ .

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