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Calculus for AP

Jon Rogawski & Ray Cannon

Chapter 7

TECHNIQUES OF INTEGRATION

Educators

MK

Problem 1

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$ $$
\int x \sin x d x ; \quad u=x, v^{\prime}=\sin x
$$

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Problem 2

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$ $$
\int x e^{2 x} d x ; \quad u=x, v^{\prime}=e^{2 x}
$$

MK
Margaret K.
Numerade Educator

Problem 3

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$ $$
\int(2 x+9) e^{x} d x ; \quad u=2 x+9, v^{\prime}=e^{x}
$$

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Problem 4

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$$$
\int x \cos 4 x d x ; \quad u=x, v^{\prime}=\cos 4 x
$$

MK
Margaret K.
Numerade Educator

Problem 5

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$$$
\int x^{3} \ln x d x ; \quad u=\ln x, v^{\prime}=x^{3}
$$

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Problem 6

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$$$
\int \tan ^{-1} x d x ; \quad u=\tan ^{-1} x, v^{\prime}=1
$$

MK
Margaret K.
Numerade Educator

Problem 7

evaluate using Integration by Parts. $$
\int(4 x-3) e^{-x} d x
$$

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Problem 8

evaluate using Integration by Parts.$$
\int(2 x+1) e^{x} d x
$$

MK
Margaret K.
Numerade Educator

Problem 9

evaluate using Integration by Parts.$$
\int x e^{5 x+2} d x
$$

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Problem 10

evaluate using Integration by Parts$$
\int x^{2} e^{x} d x
$$

MK
Margaret K.
Numerade Educator

Problem 11

evaluate using Integration by Parts$$
\int x \cos 2 x d x
$$

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Problem 12

evaluate using Integration by Parts$$
\int x \sin (3-x) d x
$$

MK
Margaret K.
Numerade Educator

Problem 13

evaluate using Integration by Parts$$
\int x^{2} \sin x d x
$$

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Problem 14

evaluate using Integration by Parts$$
\int x^{2} \cos 3 x d x
$$

MK
Margaret K.
Numerade Educator

Problem 15

evaluate using Integration by Parts$$
\int e^{-x} \sin x d x
$$

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Problem 16

evaluate using Integration by Parts$$
\int e^{x} \sin 2 x d x
$$

MK
Margaret K.
Numerade Educator

Problem 17

evaluate using Integration by Parts$$
\int e^{-5 x} \sin x d x
$$

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Problem 18

evaluate using Integration by Parts$$
\int e^{3 x} \cos 4 x d x
$$

MK
Margaret K.
Numerade Educator

Problem 19

evaluate using Integration by Parts$$
\int x \ln x d x
$$

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Problem 20

evaluate using Integration by Parts$$
\int \frac{\ln x}{x^{2}} d x
$$

MK
Margaret K.
Numerade Educator

Problem 21

evaluate using Integration by Parts$$
\int x^{2} \ln x d x
$$

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Problem 22

evaluate using Integration by Parts$$
\int x^{-5} \ln x d x
$$

MK
Margaret K.
Numerade Educator

Problem 23

evaluate using Integration by Parts$$
\int(\ln x)^{2} d x
$$

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Problem 24

evaluate using Integration by Parts$$
\int x(\ln x)^{2} d x
$$

MK
Margaret K.
Numerade Educator

Problem 25

evaluate using Integration by Parts$$
\int x \sec ^{2} x d x
$$

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Problem 26

evaluate using Integration by Parts$$
\int x \tan x \sec x d x
$$

MK
Margaret K.
Numerade Educator

Problem 27

evaluate using Integration by Parts$$
\int \cos ^{-1} x d x
$$

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Problem 28

evaluate using Integration by Parts$$
\int \sin ^{-1} x d x
$$

MK
Margaret K.
Numerade Educator

Problem 29

evaluate using Integration by Parts$$
\int \sec ^{-1} x d x
$$

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Problem 30

evaluate using Integration by Parts$$
\int x 5^{x} d x
$$

MK
Margaret K.
Numerade Educator

Problem 31

evaluate using Integration by Parts$$
\int 3^{x} \cos x d x
$$

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Problem 32

evaluate using Integration by Parts$$
\int x \sinh x d x
$$

MK
Margaret K.
Numerade Educator

Problem 33

evaluate using Integration by Parts$$
\int x^{2} \cosh x d x
$$

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Problem 34

evaluate using Integration by Parts$$
\int \cos x \cosh x d x
$$

MK
Margaret K.
Numerade Educator

Problem 35

evaluate using Integration by Parts$$
\int \tanh ^{-1} 4 x d x
$$

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Problem 36

evaluate using Integration by Parts$$
\int \sinh ^{-1} x d x
$$

MK
Margaret K.
Numerade Educator

Problem 37

evaluate using substitution and then Integration by Parts. $$\int e^{\sqrt{x}} d x$$ Hint: Let $$u=x^{1 / 2} \quad 38 . \int x^{3} e^{x^{2}} d x$$

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Problem 38

evaluate using substitution and then Integration by Parts.$$
\int x^{3} e^{x^{2}} d x
$$

MK
Margaret K.
Numerade Educator

Problem 39

evaluate using Integration by Parts, substitution, or both if necessary. $$
\int x \cos 4 x d x
$$

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Problem 40

evaluate using Integration by Parts, substitution, or both if necessary. $$
\int \frac{\ln (\ln x) d x}{x}
$$

MK
Margaret K.
Numerade Educator

Problem 41

evaluate using Integration by Parts, substitution, or both if necessary. $$
\int \frac{x d x}{\sqrt{x+1}}
$$

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Problem 42

evaluate using Integration by Parts, substitution, or both if necessary. $$
\int x^{2}\left(x^{3}+9\right)^{15} d x
$$

MK
Margaret K.
Numerade Educator

Problem 43

evaluate using Integration by Parts, substitution, or both if necessary.$$
\int \cos x \ln (\sin x) d x
$$

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Problem 44

evaluate using Integration by Parts, substitution, or both if necessary.$$
\int \sin \sqrt{x} d x
$$

MK
Margaret K.
Numerade Educator

Problem 45

evaluate using Integration by Parts, substitution, or both if necessary$$
\int \sqrt{x} e^{\sqrt{x}} d x
$$

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Problem 46

evaluate using Integration by Parts, substitution, or both if necessary$$
\int \frac{\tan \sqrt{x} d x}{\sqrt{x}}
$$

MK
Margaret K.
Numerade Educator

Problem 47

evaluate using Integration by Parts, substitution, or both if necessary$$
\int \frac{\ln (\ln x) \ln x d x}{x}
$$

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Problem 48

evaluate using Integration by Parts, substitution, or both if necessary$$
\int \sin (\ln x) d x
$$

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Problem 49

compute the definite integral. $$
\int_{0}^{3} x e^{4 x} d x
$$

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Problem 50

compute the definite integral. $$
\int_{0}^{\pi / 4} x \sin 2 x d x
$$

MK
Margaret K.
Numerade Educator

Problem 51

compute the definite integral. $$
\int_{1}^{2} x \ln x d x
$$

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Problem 52

compute the definite integral. $$
\int_{1}^{e} \frac{\ln x d x}{x^{2}}
$$

MK
Margaret K.
Numerade Educator

Problem 53

compute the definite integral. $$
\int_{0}^{\pi} e^{x} \sin x d x
$$

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Problem 54

compute the definite integral. $$
\int_{0}^{1} \tan ^{-1} x d x
$$

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Problem 55

Use Eq. (5) to evaluate $\int x^{4} e^{x} d x$

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Problem 56

Use substitution and then Eq. (5) to evaluate $\int x^{4} e^{7 x} d x$

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Problem 57

Find a reduction formula for $\int x^{n} e^{-x} d x$ similar to Eq. $(5)$

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Problem 58

Evaluate $\int x^{n} \ln x d x$ for $n \neq-1 .$ Which method should be used to evaluate $\int x^{-1} \ln x d x ?$

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Problem 59

In Exercises $59-66$ , indicate a good method for evaluating the integral
but do not evaluate). Your choices are algebraic manpulation, substitution (specify $u$ and $d u ),$ and Integration by Parts (specify u and $v^{\prime}$ ). If it appears that the techniques you have learned thus fare not suffcient, state this.
$$
\int \sqrt{x} \ln x d x
$$

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Problem 60

In Exercises $59-66$ , indicate a good method for evaluating the integral
but do not evaluate). Your choices are algebraic manpulation, substitution (specify $u$ and $d u ),$ and Integration by Parts (specify u and $v^{\prime}$ ). If it appears that the techniques you have learned thus fare not suffcient, state this.
$$
\int \frac{x^{2}-\sqrt{x}}{2 x} d x
$$

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Problem 61

In Exercises $59-66$ , indicate a good method for evaluating the integral
but do not evaluate). Your choices are algebraic manpulation, substitution (specify $u$ and $d u ),$ and Integration by Parts (specify u and $v^{\prime}$ ). If it appears that the techniques you have learned thus fare not suffcient, state this.
$$
\int \frac{x^{3} d x}{\sqrt{4-x^{2}}}
$$

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Problem 62

In Exercises $59-66$ , indicate a good method for evaluating the integral
but do not evaluate). Your choices are algebraic manpulation, substitution (specify $u$ and $d u ),$ and Integration by Parts (specify u and $v^{\prime}$ ). If it appears that the techniques you have learned thus fare not suffcient, state this.
$$
\int \frac{d x}{\sqrt{4-x^{2}}}
$$

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Problem 63

In Exercises $59-66$ , indicate a good method for evaluating the integral
but do not evaluate). Your choices are algebraic manpulation, substitution (specify $u$ and $d u ),$ and Integration by Parts (specify u and $v^{\prime}$ ). If it appears that the techniques you have learned thus fare not suffcient, state this.
$$
\int \frac{x+2}{x^{2}+4 x+3} d x
$$

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Problem 64

In Exercises $59-66$ , indicate a good method for evaluating the integral
but do not evaluate). Your choices are algebraic manpulation, substitution (specify $u$ and $d u ),$ and Integration by Parts (specify u and $v^{\prime}$ ). If it appears that the techniques you have learned thus fare not suffcient, state this.
$$
\int \frac{d x}{(x+2)\left(x^{2}+4 x+3\right)}
$$

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Problem 65

In Exercises $59-66$ , indicate a good method for evaluating the integral
but do not evaluate). Your choices are algebraic manpulation, substitution (specify $u$ and $d u ),$ and Integration by Parts (specify u and $v^{\prime}$ ). If it appears that the techniques you have learned thus fare not suffcient, state this.
$$
\int x \sin (3 x+4) d x
$$

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Problem 66

In Exercises $59-66$ , indicate a good method for evaluating the integral
but do not evaluate). Your choices are algebraic manpulation, substitution (specify $u$ and $d u ),$ and Integration by Parts (specify u and $v^{\prime}$ ). If it appears that the techniques you have learned thus fare not suffcient, state this.
$$
\int x \cos \left(9 x^{2}\right) d x
$$

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Problem 67

Evaluate $\int\left(\sin ^{-1} x\right)^{2} d x .$ Hint: Use Integration by Parts first and
then substitution.

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Problem 68

Evaluate $\int \frac{(\ln x)^{2} d x}{x^{2}} .$ Hint: Use substitution first and then Integration by Parts.

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Problem 69

Evaluate $$\int x^{7} \cos \left(x^{4}\right) d x$$

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Problem 70

Find $f(x),$ assuming that
$$\int f(x) e^{x} d x=f(x) e^{x}-\int x^{-1} e^{x} d x$$

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Problem 71

Find the volume of the solid obtained by revolving the region under $y=e^{x}$ for $0 \leq x \leq 2$ about the $y$ -axis.

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Problem 72

Find the area enclosed by $y=\ln x$ and $y=(\ln x)^{2}$

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Problem 73

Recall that the present value (PV) of an investment that pays out income continuously at a rate $R(t)$ for $T$ years is $\int_{0}^{T} R(t) e^{-r t} d t$ where $r$ is the interest rate. Find the PV if $R(t)=5000+100 t$ S/year, $r=0.05$ and $T=10$ years.

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Problem 74

Derive the reduction formula
$$\int(\ln x)^{k} d x=x(\ln x)^{k}-k \int(\ln x)^{k-1} d x$$

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Problem 75

Use Eq. (6) to calculate $\int(\ln x)^{k} d x$ for $k=2,3$

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Problem 76

Derive the reduction formulas
$$\int x^{n} \cos x d x=x^{n} \sin x-n \int x^{n-1} \sin x d x$$
$$\int x^{n} \sin x d x=-x^{n} \cos x+n \int x^{n-1} \cos x d x$$

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Problem 77

Prove that $$\int x b^{x} d x=b^{x}\left(\frac{x}{\ln b}-\frac{1}{\ln ^{2} b}\right)+C$$

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Problem 78

Define $P_{n}(x)$ by
$$\int x^{n} e^{x} d x=P_{n}(x) e^{x}+C$$ Use Eq. $(5)$ to prove that $P_{n}(x)=x^{n}-n P_{n-1}(x) .$ Use this recursion
relation to find $P_{n}(x)$ for $n=1,2,3,4$ . Note that $P_{0}(x)=1$

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Problem 79

The Integration by Parts formula can be written $$\int u(x) v(x) d x=u(x) V(x)-\int u^{\prime}(x) V(x) d x$$ where $V(x)$ satisfies $V^{\prime}(x)=v(x)$ $$
\begin{array}{l}{\text { (a) Show directly that the right-hand side of Eq. (7) does not change }} \\ {\text { if } V(x) \text { is replaced by } V(x)+C, \text { where } C \text { is a constant. }}\end{array}
$$ $$
\begin{array}{l}{\text { (b) Use } u=\tan ^{-1} x \text { and } v=x \text { in Eq. }(7) \text { to calculate } \int x \tan ^{-1} x d x} \\ {\text { but carry out the calculation twice: first with } V(x)=\frac{1}{2} x^{2} \text { and then with }} \\ {V(x)=\frac{1}{2} x^{2}+\frac{1}{2} . \text { Which choice of } V(x) \text { results in a simpler calculation? }}\end{array}
$$

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Problem 80

Prove in two ways that $$\int_{0}^{a} f(x) d x=a f(a)-\int_{0}^{a} x f^{\prime}(x) d x$$ First use Integration by Parts. Then assume $f(x)$ is increasing. Use the
substitution $u=f(x)$ to prove that $\int_{0}^{a} x f^{\prime}(x) d x$ is equal to the area
of the shaded region in Figure 1 and derive Eq. $(8)$ a second time.

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Problem 81

Assume that $f(0)=f(1)=0$ and that $f^{\prime \prime}$ exists. Prove
$$\int_{0}^{1} f^{\prime \prime}(x) f(x) d x=-\int_{0}^{1} f^{\prime}(x)^{2} d x$$ Use this to prove that if $f(0)=f(1)=0$ and $f^{\prime \prime}(x)=\lambda f(x)$ for some
constant $\lambda,$ then $\lambda<0 .$ Can you think of a function satisfying these conditions for some $\lambda$ ?

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Problem 82

Set $I(a, b)=\int_{0 } ^ {1} x ^ {a}(1-x) ^ {b} d x,$ where $a, b$ are whole numbers. $$
\begin{array}{c}{\text { (a) Use substitution to show that } I(a, b)=I(b, a) \text { . }} \\ {\text { (b) Show that } I(a, 0)=I(0, a)=\frac{1}{a+1}} \\ {\text { (c) Prove that for } a \geq 1 \text { and } b \geq 0} \\ {I(a, b)=\frac{a}{b+1} I(a-1, b+1)}\end{array}
$$ $$
\begin{array}{l}{\text { (d) Use (b) and (c) to calculate } I(1,1) \text { and } I(3,2) \text { . }} \\ {\text { (e) Show that } I(a, b)=\frac{a ! b !}{(a+b+1) !}}\end{array}
$$

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Problem 83

Let $I_{n}=\int x^{n} \cos \left(x^{2}\right) d x$ and $J_{n}=\int x^{n} \sin \left(x^{2}\right) d x$ $$
\begin{array}{l}{\text { (a) Find a reduction formula that expresses } I_{n} \text { in terms of } J_{n-2} . \text { Hint: }} \\ {\text { Write } x^{n} \cos \left(x^{2}\right) \text { as } x^{n-1}\left(x \cos \left(x^{2}\right)\right)}\end{array}
$$ $$
\begin{array}{l}{\text { (b) } \text { Use the result of (a) to show that } I_{n} \text { can be evaluated explicitly if } n \text { is odd. }}\end{array}
$$ (c) Evaluate $I_{3}$

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