University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 8

Techniques of Integration

Educators


Problem 1

Evaluate the integrals using integration by parts.
$$\int x \sin \frac{x}{2} d x$$

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Problem 2

Evaluate the integrals using integration by parts.
$$\int \theta \cos \pi \theta d \theta$$

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Problem 3

Evaluate the integrals using integration by parts.
$$\int t^{2} \cos t d t$$

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Problem 4

$$\int x^{2} \sin x d x$$

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Problem 5

Evaluate the integrals using integration by parts.
$$\int_{1}^{2} x \ln x d x$$

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Problem 6

Evaluate the integrals using integration by parts.
$$\int_{1}^{e} x^{3} \ln x d x$$

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Problem 7

Evaluate the integrals using integration by parts.
$$\int x e^{x} d x$$

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Problem 8

Evaluate the integrals using integration by parts.
$$\int x e^{3 x} d x$$

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Problem 9

Evaluate the integrals using integration by parts.
$$\int x^{2} e^{-x} d x$$

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Problem 10

Evaluate the integrals using integration by parts.
$$\int\left(x^{2}-2 x+1\right) e^{2 x} d x$$

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Problem 11

Evaluate the integrals using integration by parts.
$$\int \tan ^{-1} y d y$$

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Problem 12

Evaluate the integrals using integration by parts.
$$\int \sin ^{-1} y d y$$

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Problem 13

Evaluate the integrals using integration by parts.
$$\int x \sec ^{2} x d x$$

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Problem 14

Evaluate the integrals using integration by parts.
$$\int 4 x \sec ^{2} 2 x d x$$

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Problem 15

Evaluate the integrals using integration by parts.
$$\int x^{3} e^{x} d x$$

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Problem 16

Evaluate the integrals using integration by parts.
$$\int p^{4} e^{-p} d p$$

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Problem 17

Evaluate the integrals using integration by parts.
$$\int\left(x^{2}-5 x\right) e^{x} d x$$

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Problem 18

Evaluate the integrals using integration by parts.
$$\int\left(r^{2}+r+1\right) e^{r} d r$$

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Problem 19

Evaluate the integrals using integration by parts.
$$\int x^{5} e^{x} d x$$

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Problem 20

Evaluate the integrals using integration by parts.
$$\int t^{2} e^{4 t} d t$$

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Problem 21

Evaluate the integrals using integration by parts.
$$\int e^{\theta} \sin \theta d \theta$$

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Problem 22

Evaluate the integrals using integration by parts.
$$\int e^{\theta} \sin \theta d \theta$$

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Problem 23

Evaluate the integrals using integration by parts.
$$\int e^{2 x} \cos 3 x d x$$

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Problem 24

Evaluate the integrals using integration by parts.
$$\int e^{-2 x} \sin 2 x d x$$

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Problem 25

Evaluate the integrals by using a substitution prior to integration by parts.
$$\int e^{\sqrt{3 x+9}} d s$$

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Problem 26

Evaluate the integrals by using a substitution prior to integration by parts.
$$\int_{0}^{1} x \sqrt{1-x} d x$$

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Problem 27

Evaluate the integrals by using a substitution prior to integration by parts.
$$\int_{0}^{\pi / 3} x \tan ^{2} x d x$$

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Problem 28

Evaluate the integrals by using a substitution prior to integration by parts.
$$\int \ln \left(x+x^{2}\right) d x$$

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Problem 29

Evaluate the integrals by using a substitution prior to integration by parts.
$$\int z(\ln z)^{2} d z$$

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Problem 30

Evaluate the integrals by using a substitution prior to integration by parts.
$$\int \sin (\ln x) d x$$

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Problem 31

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int x \sec x^{2} d x$$

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Problem 32

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$$

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Problem 33

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int x(\ln x)^{2} d x$$

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Problem 34

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \frac{1}{x(\ln x)^{2}} d x$$

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Problem 35

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \frac{\ln x}{x^{2}} d x$$

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Problem 36

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \frac{(\ln x)^{3}}{x} d x$$

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Problem 37

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int x^{3} e^{x^{4}} d x$$

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Problem 38

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int x^{5} e^{x^{3}} d x$$

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Problem 39

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int x^{3} \sqrt{x^{2}+1} d x$$

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Problem 40

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int x^{2} \sin x^{3} d x$$

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Problem 41

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \sin 3 x \cos 2 x \, d x$$

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Problem 42

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \sin 2 x \cos 4 x \, d x$$

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Problem 43

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \sqrt{x} \ln x \, d x$$

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Problem 44

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

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Problem 45

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \cos \sqrt{x} d x$$

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Problem 46

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \sqrt{x} e^{\sqrt{x}} d x$$

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Problem 47

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int_{0}^{\pi / 2} \theta^{2} \sin 2 \theta d \theta$$

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Problem 48

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int_{0}^{\pi / 2} x^{3} \cos 2 x d x$$

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Problem 49

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int_{2 / \sqrt{3}}^{2} t \sec ^{-1} t d t$$

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Problem 50

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int_{0}^{1 / \sqrt{2}} 2 x \sin ^{-1}\left(x^{2}\right) d x$$

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Problem 51

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int x \tan ^{-1} x d x$$

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Problem 52

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int x^{2} \tan ^{-1} \frac{x}{2} d x$$

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Problem 53

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int\left(1+2 x^{2}\right) e^{x^{2}} d x$$

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Problem 54

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \frac{x e^{x}}{(x+1)^{2}} d x$$

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Problem 55

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \sqrt{x}\left(\sin ^{-1} \sqrt{x}\right) d x$$

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Problem 56

Evaluate the integrals. Some integrals do not require integration by parts.
$$\int \frac{\left(\sin ^{-1} x\right)^{2}}{\sqrt{1-x^{2}}} d x$$

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Problem 57

Finding area Find the area of the region enclosed by the curve $y=x \sin x$ and the $x$ -axis (see the accompanying figure) for
a. $0 \leq x \leq \pi$
b. $\pi \leq x \leq 2 \pi$
c. $2 \pi=r=3 \pi$
d. What pattern do you see here? What is the area between the curve and the $x$ -axis for $n \pi \leq x \leq(n+1) \pi, n$ an arbitrary nonnegative integer? Give reasons for your answer.
(FIGURE CAN'T COPY)

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Problem 58

Finding area Find the area of the region enclosed by the curve $y=x \cos x$ and the $x$ -axis (see the accompanying figure) for
a. $\pi / 2 \leq x \leq 3 \pi / 2$
b. $3 \pi / 2 \leq x \leq 5 \pi / 2$
c. $5 \pi / 2 \leq x \leq 7 \pi / 2$
d. What pattern do you see? What is the area between the curve and the $x$ -axis for
$$\left(\frac{2 n-1}{2}\right) \pi \leq x \leq\left(\frac{2 n+1}{2}\right) \pi$$
$n$ an arbitrary positive integer? Give reasons for your answer.
(FIGURE CAN'T COPY)

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Problem 59

Finding volume Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y=e^{x},$ and the line $x=\ln 2$ about the line $x=\ln 2$

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Problem 60

Finding volume Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y=e^{-x},$ and the line $x=1$
a. about the $y$ -axis.
b. about the line $x=1$

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Problem 61

Finding volume Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve $y=\cos x, 0 \leq x \leq \pi / 2,$ about
a. the $y$ -axis.
b. the line $x=\pi / 2$

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Problem 62

Finding volume Find the volume of the solid generated roy revolving the region bounded by the $x$ -axis and the curve $y=x \sin x, 0 \leq x \leq \pi,$ about
a. the $y$ -axis.
b. the line $x=\pi$
(FIGURE CAN'T COPY)

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Problem 63

Consider the region bounded by the graphs of $y=\ln x, y=0$ and $x=e$
a. Find the area of the region.
b. Find the volume of the solid formed by revolving this region about the $x$ -axis.
c. Find the volume of the solid formed by revolving this region about the line $x=-2$
d. Find the centroid of the region.

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Problem 64

Consider the region bounded by the graphs of $y=\tan ^{-1} x, y=0$ and $x=1$
a. Find the area of the region.
b. Find the volume of the solid formed by revolving this region about the $y$ -axis.

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Problem 65

Average value $\quad$ A retarding force, symbolized by the dashpot in the accompanying figure, slows the motion of the weighted spring so that the mass's position at time $t$ is
$$y=2 e^{-t} \cos t, \quad t \geq 0$$
Find the average value of $y$ over the interval $0 \leq t \leq 2 \pi$
(FIGURE CAN'T COPY)

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Problem 66

Average value In a mass-spring-dashpot system like the one in Exercise $65,$ the mass's position at time $t$ is
$$y=4 e^{-t}(\sin t-\cos t), \quad t \geq 0$$
Find the average value of $y$ over the interval $0 \leq t \leq 2 \pi$

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Problem 67

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

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Problem 68

Use integration by parts to establish the reduction formula.
$$\int x^{n} \sin x d x=-x^{n} \cos x+n \int x^{n-1} \cos x d x$$

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Problem 69

Use integration by parts to establish the reduction formula.
$$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, a \neq 0$$

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Problem 70

Use integration by parts to establish 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 71

Use integration by parts to establish the reduction formula.
$$\begin{aligned}
\int x^{m}(\ln x)^{n} d x=& \frac{x^{m+1}}{m+1}(\ln x)^{n} \\
&-\frac{n}{m+1} \int x^{m}(\ln x)^{n-1} d x, m \neq-1
\end{aligned}$$

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Problem 72

Use integration by parts to establish the reduction formula.
$$\begin{aligned}
\int x^{n} \sqrt{x+1} d x &=\frac{2 x^{n}}{2 n+3}(x+1)^{3 / 2} \\
&-\frac{2 n}{2 n+3} \int x^{n-1} \sqrt{x+1} d x
\end{aligned}$$

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Problem 73

Use integration by parts to establish the reduction formula.
$$\begin{aligned}
\int \frac{x^{n}}{\sqrt{x+1}} d x=& \frac{2 x^{n}}{2 n+1} \sqrt{x+1} \\
&-\frac{2 n}{2 n+1} \int \frac{x^{n-1}}{\sqrt{x+1}} d x
\end{aligned}$$

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Problem 74

Use Example 5 to show that
$$\begin{aligned}
\int_{0}^{\pi / 2} \sin ^{n} x d x &=\int_{0}^{\pi / 2} \cos ^{n} x d x \\
&=\left\{\begin{array}{l}
\left(\frac{\pi}{2}\right) \frac{1 \cdot 3 \cdot 5 \cdot \cdots(n-1)}{2 \cdot 4 \cdot 6 \cdots n}, n \text { even } \\
\frac{2 \cdot 4 \cdot 6 \cdots(n-1)}{1 \cdot 3 \cdot 5 \cdots n}, n \text { odd }
\end{array}\right.
\end{aligned}$$

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Problem 75

Show that
$$\int_{a}^{b}\left(\int_{x}^{b} f(t) d t\right) d x=\int_{a}^{b}(x-a) f(x) d x$$

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Problem 76

Use integration by parts to obtain the formula
$$\int \sqrt{1-x^{2}} d x=\frac{1}{2} x \sqrt{1-x^{2}}+\frac{1}{2} \int \frac{1}{\sqrt{1-x^{2}}} d x$$

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Problem 77

Use the formula
$$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y$$
to evaluate the integrals .Express your answers in terms of $x$
$$\int \sin ^{-1} x d x$$

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Problem 78

Use the formula
$$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y$$
to evaluate the integrals .Express your answers in terms of $x$
$$\int \tan ^{-1} x d x$$

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Problem 79

Use the formula
$$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y$$
to evaluate the integrals .Express your answers in terms of $x$
$$\int \sec ^{-1} x d x$$

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Problem 80

Use the formula
$$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y$$
to evaluate the integrals .Express your answers in terms of $x$
$$\int \log _{2} x d x$$

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Problem 81

Compare the results of using Equations (4) and (5)
Equations (4) and (5) give different formulas for the integral of $=0 \mathrm{s}^{-1} x$
$$\text { a. } \int \cos ^{-1} x d x=x \cos ^{-1} x-\sin \left(\cos ^{-1} x\right)+C$$
$$\text { b. } \int \cos ^{-1} x d x=x \cos ^{-1} x-\sqrt{1-x^{2}}+C$$
Can both integrations be correct? Explain.

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Problem 82

Compare the results of using Equations (4) and (5)
Equations (4) and (5) lead to different formulas for the integral of $\tan ^{-1} x$
$$\text { a. } \int \tan ^{-1} x d x=x \tan ^{-1} x-\ln \sec \left(\tan ^{-1} x\right)+C$$
$$\text { b. } \int \tan ^{-1} x d x=x \tan ^{-1} x-\ln \sqrt{1+x^{2}}+C$$
Can both integrations be correct? Explain.

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Problem 83

Evaluate the integrals with (a) $\mathrm{Eq} .(4)$ and $(\mathrm{b})$ Eq. (5). In each case, check your work by differentiating your answer with respect to $x$.
$$\int \sinh ^{-1} x d x$$

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Problem 84

Evaluate the integrals with (a) $\mathrm{Eq} .(4)$ and $(\mathrm{b})$ Eq. (5). In each case, check your work by differentiating your answer with respect to $x$.
$$\int \tanh ^{-1} x d x$$

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