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Thomas Calculus 12

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Chapter 8

Techniques of Integration

Educators


Problem 1

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int x \sin \frac{x}{2} d x$$

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Problem 2

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int \theta \cos \pi \theta d \theta$$

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Problem 3

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int t^{2} \cos t d t$$

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Problem 4

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int x^{2} \sin x d x$$

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Problem 5

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int_{1}^{2} x \ln x d x$$

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Problem 6

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int_{1}^{e} x^{3} \ln x d x$$

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Problem 7

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int x e^{x} d x$$

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Problem 8

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int x e^{3 x} d x$$

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Problem 9

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int x^{2} e^{-x} d x$$

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Problem 10

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\frac{1}{2}\left(x^{2}-2 x+1\right) e^{2 x}-\frac{1}{2}(x-1) e^{2 x}+\frac{1}{4} e^{2 x}+C$$

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Problem 11

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int \tan ^{-1} y d y$$

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Problem 12

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int \sin ^{-1} y d y$$

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Problem 13

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int x \sec ^{2} x d x$$

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Problem 14

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int 4 x \sec ^{2} 2 x d x$$

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Problem 15

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int x^{3} e^{x} d x$$

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Problem 16

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int p^{4} e^{-p} d p$$

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Problem 17

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int\left(x^{2}-5 x\right) e^{x} d x$$

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Problem 18

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int\left(r^{2}+r+1\right) e^{r} d r$$

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Problem 19

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int x^{5} e^{x} d x$$

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Problem 20

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int t^{2} e^{4 t} d t$$

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Problem 21

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int e^{\theta} \sin \theta d \theta$$

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Problem 22

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int e^{-y} \cos y d y$$

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Problem 23

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int e^{2 x} \cos 3 x d x$$

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Problem 24

Evaluate the integrals in Exercises $1-24$ using integration by parts.
$$\int e^{-2 x} \sin 2 x d x$$

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Problem 25

Evaluate the integrals in Exercises $25-30$ by using a substitution prior to integration by parts.
$$\int e^{\sqrt{3 s+9}} d s$$

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Problem 26

Evaluate the integrals in Exercises $25-30$ 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 in Exercises $25-30$ 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 in Exercises $25-30$ 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 in Exercises $25-30$ by using a substitution prior to integration by parts.
$$\int \sin (\ln x) d x$$

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Problem 30

Evaluate the integrals in Exercises $25-30$ by using a substitution prior to integration by parts.
$$\int z(\ln z)^{2} d z$$

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Problem 31

Evaluate the integrals in Exercises $31-50 .$ Some integrals do not require integration by parts.
$$\int x \sec x^{2} d x$$

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Problem 32

Evaluate the integrals in Exercises $31-50 .$ 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 in Exercises $31-50 .$ Some integrals do not require integration by parts.
$$\int x(\ln x)^{2} d x$$

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Problem 34

Evaluate the integrals in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ Some integrals do not require integration by parts.
$$\int e^{x} \sin e^{x} d x$$

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Problem 44

Evaluate the integrals in Exercises $31-50 .$ 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 in Exercises $31-50 .$ Some integrals do not require integration by parts.
$$\int \cos \sqrt{x} d x$$

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Problem 46

Evaluate the integrals in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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 in Exercises $31-50 .$ 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

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. $2 \pi \leq x \leq 3 \pi$
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.

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Problem 52

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.

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Problem 53

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 54

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 55

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 56

Finding volume Find the volume of the solid generated by revolving the region bounded by the $x$ -axis and the curve $y=x \sin x, 0 \leq x \leq \pi,$ about
a. a. the $y$ -axis.
b. the line $x=\pi$ .
(See Exercise 51 for a graph.)

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Problem 57

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
c. Find the volume of the solid formed by revolving this region
a. Find the line $x=-2 .$
d. Find the centroid of the region.

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Problem 58

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 59

Average value 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 $0 \leq t \leq 2 \pi$

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Problem 60

Average value In a mass-spring-dashpot system like the one in Exercise $59,$ 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 61

In Exercises $61-64,$ 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 62

In Exercises $61-64,$ 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 63

In Exercises $61-64,$ use integration by parts to establish the reduction formula.
$$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \quad a \neq 0$$

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Problem 64

In Exercises $61-64,$ 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 65

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 66

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 67

Integration by parts leads to a rule for integrating inverses that usually gives good results:
$$\begin{aligned} \int f^{-1}(x) d x &=\int y f^{\prime}(y) d y \quad \quad y=f^{-1}(x), \quad x=f(y) \\ &=y f(y)-\int f(y) d y \quad u=y, d v=f^{\prime}(y) d y \\ &=x f^{-1}(x)-\int f(y) d y \end{aligned}$$
The idea is to take the most complicated part of the integral, in this case $f^{-1}(x),$ and simplify it first. For the integral of $\ln x,$ we get
$$\begin{aligned} \int \ln x d x &=\int y e^{y} d y \\ &=y e^{y}-e^{y}+C \\ &=x \ln x-x+C \end{aligned} \quad \begin{aligned} y &=\ln x, \quad x=e^{y} \\ d x &=e^{y} d y \end{aligned}$$
For the integral of $\cos ^{-1} x$ we get
$$\begin{aligned} \int \cos ^{-1} x d x &=x \cos ^{-1} x-\int \cos y d y \\ &=x \cos ^{-1} x-\sin y+C \\ &=x \cos ^{-1} x-\sin \left(\cos ^{-1} x\right)+C \end{aligned}$$
Use the formula
$$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y \quad y=f^{-1}(x)$$
to evaluate the integrals in Exercises $67-70 .$ Express your answers in terms of $x .$
$$\int \sin ^{-1} x d x$$

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Problem 68

Integration by parts leads to a rule for integrating inverses that usually gives good results:
$$\begin{aligned} \int f^{-1}(x) d x &=\int y f^{\prime}(y) d y \quad \quad y=f^{-1}(x), \quad x=f(y) \\ &=y f(y)-\int f(y) d y \quad u=y, d v=f^{\prime}(y) d y \\ &=x f^{-1}(x)-\int f(y) d y \end{aligned}$$
The idea is to take the most complicated part of the integral, in this case $f^{-1}(x),$ and simplify it first. For the integral of $\ln x,$ we get
$$\begin{aligned} \int \ln x d x &=\int y e^{y} d y \\ &=y e^{y}-e^{y}+C \\ &=x \ln x-x+C \end{aligned} \quad \begin{aligned} y &=\ln x, \quad x=e^{y} \\ d x &=e^{y} d y \end{aligned}$$
For the integral of $\cos ^{-1} x$ we get
$$\begin{aligned} \int \cos ^{-1} x d x &=x \cos ^{-1} x-\int \cos y d y \\ &=x \cos ^{-1} x-\sin y+C \\ &=x \cos ^{-1} x-\sin \left(\cos ^{-1} x\right)+C \end{aligned}$$
Use the formula
$$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y \quad y=f^{-1}(x)$$
to evaluate the integrals in Exercises $67-70 .$ Express your answers in terms of $x .$
$$\int \tan ^{-1} x d x$$

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Problem 69

Integration by parts leads to a rule for integrating inverses that usually gives good results:
$$\begin{aligned} \int f^{-1}(x) d x &=\int y f^{\prime}(y) d y \quad \quad y=f^{-1}(x), \quad x=f(y) \\ &=y f(y)-\int f(y) d y \quad u=y, d v=f^{\prime}(y) d y \\ &=x f^{-1}(x)-\int f(y) d y \end{aligned}$$
The idea is to take the most complicated part of the integral, in this case $f^{-1}(x),$ and simplify it first. For the integral of $\ln x,$ we get
$$\begin{aligned} \int \ln x d x &=\int y e^{y} d y \\ &=y e^{y}-e^{y}+C \\ &=x \ln x-x+C \end{aligned} \quad \begin{aligned} y &=\ln x, \quad x=e^{y} \\ d x &=e^{y} d y \end{aligned}$$
For the integral of $\cos ^{-1} x$ we get
$$\begin{aligned} \int \cos ^{-1} x d x &=x \cos ^{-1} x-\int \cos y d y \\ &=x \cos ^{-1} x-\sin y+C \\ &=x \cos ^{-1} x-\sin \left(\cos ^{-1} x\right)+C \end{aligned}$$
Use the formula
$$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y \quad y=f^{-1}(x)$$
to evaluate the integrals in Exercises $67-70 .$ Express your answers in terms of $x .$
$$\int \sec ^{-1} x d x$$

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Problem 70

Integration by parts leads to a rule for integrating inverses that usually gives good results:
$$\begin{aligned} \int f^{-1}(x) d x &=\int y f^{\prime}(y) d y \quad \quad y=f^{-1}(x), \quad x=f(y) \\ &=y f(y)-\int f(y) d y \quad u=y, d v=f^{\prime}(y) d y \\ &=x f^{-1}(x)-\int f(y) d y \end{aligned}$$
The idea is to take the most complicated part of the integral, in this case $f^{-1}(x),$ and simplify it first. For the integral of $\ln x,$ we get
$$\begin{aligned} \int \ln x d x &=\int y e^{y} d y \\ &=y e^{y}-e^{y}+C \\ &=x \ln x-x+C \end{aligned} \quad \begin{aligned} y &=\ln x, \quad x=e^{y} \\ d x &=e^{y} d y \end{aligned}$$
For the integral of $\cos ^{-1} x$ we get
$$\begin{aligned} \int \cos ^{-1} x d x &=x \cos ^{-1} x-\int \cos y d y \\ &=x \cos ^{-1} x-\sin y+C \\ &=x \cos ^{-1} x-\sin \left(\cos ^{-1} x\right)+C \end{aligned}$$
Use the formula
$$\int f^{-1}(x) d x=x f^{-1}(x)-\int f(y) d y \quad y=f^{-1}(x)$$
to evaluate the integrals in Exercises $67-70 .$ Express your answers in terms of $x .$
$$\int \log _{2} x d x$$
Another way to integrate $f^{-1}(x)$ (when $f^{-1}$ is integrable, of course) is to use integration by parts with $u=f^{-1}(x)$ and $d v=d x$ to rewrite the integral of $f^{-1}$ as
$$\int f^{-1}(x) d x=x f^{-1}(x)-\int x\left(\frac{d}{d x} f^{-1}(x)\right) d x$$

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Problem 71

Exercises 71 and 72 compare the results of using Equations $(4)$ and $(5)$
Equations $(4)$ and $(5)$ give different formulas for the integral of $\cos ^{-1} x :$
a. $\int \cos ^{-1} x d x=x \cos ^{-1} x-\sin \left(\cos ^{-1} x\right)+C$
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 72

Equations $(4)$ and $(5)$ lead to different formulas for the integral of $\tan ^{-1} x :$
a. $\int \tan ^{-1} x d x=x \tan ^{-1} x-\ln \sec \left(\tan ^{-1} x\right)+C$
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 73

Evaluate the integrals in Exercises 73 and 74 with (a) Eq. (4) and (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 74

Evaluate the integrals in Exercises 73 and 74 with (a) Eq. (4) and (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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