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Calculus 6th

Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason

Chapter 7

Integration

Educators


Problem 1

Use substitution to express each of the following integrals as a multiple of $\int_{a}^{b}(1 / w) d w$ for some $a$ and $b$ Then evaluate the integrals.
(a) $\int_{0}^{1} \frac{x}{1+x^{2}} d x$
(b) $\int_{0}^{\pi / 4} \frac{\sin x}{\cos x} d x$

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Problem 2

(a) Find the derivatives of $\sin \left(x^{2}+1\right)$ and $\sin \left(x^{3}+1\right)$
(b) Use your answer to part (a) to find antiderivatives of:
(i) $x \cos \left(x^{2}+1\right)$
(ii) $x^{2} \cos \left(x^{3}+1\right)$
(c) Find the general antiderivatives of:
(i) $x \sin \left(x^{2}+1\right)$
(ii) $x^{2} \sin \left(x^{3}+1\right)$

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Problem 3

Find the integrals. Check your answers by differentiation.
$$\int e^{3 x} d x$$

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Problem 4

Find the integrals. Check your answers by differentiation.
$$\int t e^{t^{2}} d t$$

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Problem 5

Find the integrals. Check your answers by differentiation.
$$\int e^{-x} d x$$

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Problem 6

Find the integrals. Check your answers by differentiation.
$$\int 25 e^{-0.2 t} d t$$

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Problem 7

Find the integrals. Check your answers by differentiation.
$$\int \sin (2 x) d x$$

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Problem 8

Find the integrals. Check your answers by differentiation.
$$\int t \cos \left(t^{2}\right) d t$$

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Problem 9

Find the integrals. Check your answers by differentiation.
$$\int \sin (3-t) d t$$

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Problem 10

Find the integrals. Check your answers by differentiation.
$$\int x e^{-x^{2}} d x$$

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Problem 11

Find the integrals. Check your answers by differentiation.
$$\int(r+1)^{3} d r$$

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Problem 12

Find the integrals. Check your answers by differentiation.
$$\int y\left(y^{2}+5\right)^{8} d y$$

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Problem 13

Find the integrals. Check your answers by differentiation.
$$\int x^{2}\left(1+2 x^{3}\right)^{2} d x$$

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Problem 14

Find the integrals. Check your answers by differentiation.
$$\int t^{2}\left(t^{3}-3\right)^{10} d t$$

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Problem 15

Find the integrals. Check your answers by differentiation.
$$\int x\left(x^{2}+3\right)^{2} d x$$

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Problem 16

Find the integrals. Check your answers by differentiation.
$$\int x\left(x^{2}-4\right)^{7 / 2} d x$$

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Problem 17

Find the integrals. Check your answers by differentiation.
$$-\int y^{2}(1+y)^{2} d y$$

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Problem 18

Find the integrals. Check your answers by differentiation.
$$\int(2 t-7)^{73} d t$$

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Problem 19

Find the integrals. Check your answers by differentiation.
$$\int x^{2} e^{x^{3}+1} d x$$

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Problem 20

Find the integrals. Check your answers by differentiation.
$$\int \frac{d y}{y+5}$$

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Problem 21

Find the integrals. Check your answers by differentiation.
$$\int \frac{1}{\sqrt{4-x}} d x$$

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Problem 22

Find the integrals. Check your answers by differentiation.
$$\int\left(x^{2}+3\right)^{2} d x$$

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Problem 23

Find the integrals. Check your answers by differentiation.
$$\int \sin \theta(\cos \theta+5)^{7} d \theta$$

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Problem 24

Find the integrals. Check your answers by differentiation.
$$\int \sqrt{\cos 3 t} \sin 3 t \, d t$$

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Problem 25

Find the integrals. Check your answers by differentiation.
$$\int \sin ^{6} \theta \cos \theta d \theta$$

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Problem 26

Find the integrals. Check your answers by differentiation.
$$\int \sin ^{3} \alpha \cos \alpha d \alpha$$

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Problem 27

Find the integrals. Check your answers by differentiation.
$$\int \sin ^{6}(5 \theta) \cos (5 \theta) d \theta$$

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Problem 28

Find the integrals. Check your answers by differentiation.
$$\int \tan (2 x) d x$$

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Problem 29

Find the integrals. Check your answers by differentiation.
$$\int \frac{(\ln z)^{2}}{z} d z$$

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Problem 30

Find the integrals. Check your answers by differentiation.
$$\int \frac{e^{t}+1}{e^{t}+t} d t$$

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Problem 31

Find the integrals. Check your answers by differentiation.
$$\int \frac{(t+1)^{2}}{t^{2}} d t$$

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Problem 32

Find the integrals. Check your answers by differentiation.
$$\int \frac{y}{y^{2}+4} d y$$

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Problem 33

Find the integrals. Check your answers by differentiation.
$$\int \frac{d x}{1+2 x^{2}}$$

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Problem 34

Find the integrals. Check your answers by differentiation.
$$\int \frac{d x}{\sqrt{1-4 x^{2}}}$$

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Problem 35

Find the integrals. Check your answers by differentiation.
$$\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x$$

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Problem 36

Find the integrals. Check your answers by differentiation.
$$\int \frac{e^{\sqrt{y}}}{\sqrt{y}} d y$$

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Problem 37

Find the integrals. Check your answers by differentiation.
$$\int \frac{1+e^{x}}{\sqrt{x+e^{x}}} d x$$

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Problem 38

Find the integrals. Check your answers by differentiation.
$$\int \frac{e^{x}}{2+e^{x}} d x$$

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Problem 39

Find the integrals. Check your answers by differentiation.
$$\int \frac{x+1}{x^{2}+2 x+19} d x$$

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Problem 40

Find the integrals. Check your answers by differentiation.
$$\int \frac{t}{1+3 t^{2}} d t$$

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Problem 41

Find the integrals. Check your answers by differentiation.
$$\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x$$

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Problem 42

Find the integrals. Check your answers by differentiation.
$$\int \frac{x \cos \left(x^{2}\right)}{\sqrt{\sin \left(x^{2}\right)}} d x$$

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Problem 43

Find the integrals. Check your answers by differentiation.
$$\int \sinh 3 t \, d t$$

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Problem 44

Find the integrals. Check your answers by differentiation.
$$\int \cosh x \, d x$$

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Problem 45

Find the integrals. Check your answers by differentiation.
$$\int \cosh (2 w+1) d w$$

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Problem 46

Find the integrals. Check your answers by differentiation.
$$\int(\sinh z) e^{\cosh z} d z$$

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Problem 47

Find the integrals. Check your answers by differentiation.
$$\int \cosh ^{2} x \sinh x d x$$

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Problem 48

Find the integrals. Check your answers by differentiation.
$$\int x \cosh x^{2} d x$$

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Problem 49

Find the general antiderivative. Check your answers by differentiation.
$$p(t)=\pi t^{3}+4 t$$

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Problem 50

Find the general antiderivative. Check your answers by differentiation.
$$f(x)=\sin 3 x$$

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Problem 51

Find the general antiderivative. Check your answers by differentiation.
$$f(x)=2 x \cos \left(x^{2}\right)$$

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Problem 52

Find the general antiderivative. Check your answers by differentiation.
$$r(t)=12 t^{2} \cos \left(t^{3}\right)$$

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Problem 53

Find the general antiderivative. Check your answers by differentiation.
$$f(x)=\sin (2-5 x)$$

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Problem 54

Find the general antiderivative. Check your answers by differentiation.
$$f(x)=e^{\sin x} \cos x$$

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Problem 55

Find the general antiderivative. Check your answers by differentiation.
$$f(x)=\frac{x}{x^{2}+1}$$

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Problem 56

Find the general antiderivative. Check your answers by differentiation.
$$f(x)=\frac{1}{3 \cos ^{2}(2 x)}$$

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Problem 57

Use the Fundamental Theorem to calculate the definite integrals.
$$\int_{0}^{\pi} \cos (x+\pi) d x$$

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Problem 58

Use the Fundamental Theorem to calculate the definite integrals.
$$\int_{0}^{1 / 2} \cos (\pi x) d x$$

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Problem 59

Use the Fundamental Theorem to calculate the definite integrals.
$$\int_{0}^{\pi / 2} e^{-\cos \theta} \sin \theta d \theta$$

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Problem 60

Use the Fundamental Theorem to calculate the definite integrals.
$$\int_{1}^{2} 2 x e^{x^{2}} d x$$

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Problem 61

Use the Fundamental Theorem to calculate the definite integrals.
$$\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

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Problem 62

Use the Fundamental Theorem to calculate the definite integrals.
$$\int_{-1}^{e-2} \frac{1}{t+2} d t$$

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Problem 63

Use the Fundamental Theorem to calculate the definite integrals.
$$\int_{1}^{4} \frac{\cos \sqrt{x}}{\sqrt{x}} d x$$

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Problem 64

Use the Fundamental Theorem to calculate the definite integrals.
$$\int_{0}^{2} \frac{x}{\left(1+x^{2}\right)^{2}} d x$$

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Problem 65

Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.
$$\int_{-1}^{3}\left(x^{3}+5 x\right) d x$$

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Problem 66

Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.
$$\int_{-1}^{1} \frac{1}{1+y^{2}} d y$$

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Problem 67

Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.
$$\int_{1}^{3} \frac{1}{x} d x$$

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Problem 68

Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.
$$\int_{1}^{3} \frac{d t}{(t+7)^{2}}$$

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Problem 69

Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.

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Problem 70

Evaluate the definite integrals. Whenever possible, use the Fundamental Theorem of Calculus, perhaps after a substitution. Otherwise, use numerical methods.
$$\int_{1}^{2} \frac{\sin t}{t} d t$$

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Problem 71

Find the integrals.
$$\int y \sqrt{y+1} d y$$

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Problem 72

Find the integrals.
$$\int z(z+1)^{1 / 3} d z$$

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Problem 73

Find the integrals.
$$\int \frac{t^{2}+t}{\sqrt{t+1}} d t$$

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Problem 74

Find the integrals.
$$\int \frac{d x}{2+2 \sqrt{x}}$$

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Problem 75

Find the integrals.
$$\int x^{2} \sqrt{x-2} d x$$

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Problem 76

Find the integrals.
$$\int(z+2) \sqrt{1-z} d z$$

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Problem 77

Find the integrals.
$$\int \frac{t}{\sqrt{t+1}} d t$$

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Problem 78

Find the integrals.
$$\int \frac{3 x-2}{\sqrt{2 x+1}} d x$$

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Problem 79

Show the two integrals are equal using a substitution.
$$\int_{0}^{\pi / 3} 3 \sin ^{2}(3 x) d x=\int_{0}^{\pi} \sin ^{2}(y) d y$$

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Problem 80

Show the two integrals are equal using a substitution.
$$\int_{1}^{2} 2 \ln \left(s^{2}+1\right) d s=\int_{1}^{4} \frac{\ln (t+1)}{\sqrt{t}} d t$$

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Problem 81

Show the two integrals are equal using a substitution.
$$\int_{1}^{e}(\ln w)^{3} d w=\int_{0}^{1} z^{3} e^{z} d z$$

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Problem 82

Show the two integrals are equal using a substitution.
$$\int_{0}^{\pi} x \cos (\pi-x) d x=\int_{0}^{\pi}(\pi-t) \cos t d t$$

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Problem 83

Using the substitution $w=x^{2},$ find a function $g(w)$ such that $\int_{\sqrt{a}}^{\sqrt{b}} d x=\int_{a}^{b} g(w) d w$ for all $0<a<b$.

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Problem 84

Using the substitution $w=e^{x},$ find a function $g(w)$ such that $\int_{a}^{b} e^{-x} d x=\int_{e^{a}}^{e^{b}} g(w) d w$ for all $a<b$.

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Problem 85

Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals.
$$\int \frac{e^{x} d x}{1+e^{2 x}} \text { and } \int \frac{\cos x d x}{1+\sin ^{2} x}$$

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Problem 86

Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals.
$$\int \frac{\ln x}{x} d x \text { and } \, \int x \, d x$$

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Problem 87

Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals.
$$\int e^{\sin x} \cos x d x \quad \text { and } \quad \int \frac{e^{\arcsin x}}{\sqrt{1-x^{2}}} d x$$

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Problem 88

Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals.
$$\int(\sin x)^{3} \cos x d x \quad \text { and } \quad \int\left(x^{3}+1\right)^{3} x^{2} d x$$

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Problem 89

Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals.
$$\int \sqrt{x+1} d x \text { and } \, \int \frac{\sqrt{1+\sqrt{x}}}{\sqrt{x}} d x$$

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Problem 90

Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:
$$\begin{array}{c|c|c|r|r|r}
\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\
\hline f(x) & 5 & 7 & 8 & 10 & 11 \\
\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\
\hline
\end{array}$$
$$\int_{0}^{1} f^{\prime}(x) \sin f(x) d x$$

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Problem 91

Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:
$$\begin{array}{c|c|c|r|r|r}
\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\
\hline f(x) & 5 & 7 & 8 & 10 & 11 \\
\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\
\hline
\end{array}$$
$$\int_{1}^{3} f^{\prime}(x) e^{f(x)} d x$$

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Problem 92

Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:
$$\begin{array}{c|c|c|r|r|r}
\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\
\hline f(x) & 5 & 7 & 8 & 10 & 11 \\
\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\
\hline
\end{array}$$
$$\int_{1}^{3} \frac{f^{\prime}(x)}{f(x)} d x$$

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Problem 93

Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:
$$\begin{array}{c|c|c|r|r|r}
\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\
\hline f(x) & 5 & 7 & 8 & 10 & 11 \\
\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\
\hline
\end{array}$$
$$\int_{0}^{1} e^{x} f^{\prime}\left(e^{x}\right) d x$$

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Problem 94

Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:
$$\begin{array}{c|c|c|r|r|r}
\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\
\hline f(x) & 5 & 7 & 8 & 10 & 11 \\
\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\
\hline
\end{array}$$
$$\int_{1}^{e} \frac{f^{\prime}(\ln x)}{x} d x$$

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Problem 95

Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:
$$\begin{array}{c|c|c|r|r|r}
\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\
\hline f(x) & 5 & 7 & 8 & 10 & 11 \\
\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\
\hline
\end{array}$$
$$\int_{0}^{1} f^{\prime}(x)(f(x))^{2} d x$$

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Problem 96

Evaluate the integral. Your answer should not contain $f,$ which is a differentiable function with the following values:
$$\begin{array}{c|c|c|r|r|r}
\hline x & 0 & 1 & \pi / 2 & \mathrm{e} & 3 \\
\hline f(x) & 5 & 7 & 8 & 10 & 11 \\
\hline f^{\prime}(x) & 2 & 4 & 6 & 9 & 12 \\
\hline
\end{array}$$
$$\int_{0}^{\pi / 2} \sin x \cdot f^{\prime}(\cos x) d x$$

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Problem 97

Find an expression for the integral which contains $g$ but no integral sign.
$$\int g^{\prime}(x)(g(x))^{4} d x$$

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Problem 98

Find an expression for the integral which contains $g$ but no integral sign.
$$\int g^{\prime}(x) e^{g(x)} d x$$

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Problem 99

Find an expression for the integral which contains $g$ but no integral sign.
$$\int g^{\prime}(x) \sin g(x) d x$$

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Problem 100

Find an expression for the integral which contains $g$ but no integral sign.
$$\int g^{\prime}(x) \sqrt{1+g(x)} d x$$

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Problem 101

Find a substitution $w$ and constants $k, n$ so that the integral has the form $\int k w^{n} d w$.
$$\int x^{2} \sqrt{1-4 x^{3}} d x$$

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Problem 102

Find a substitution $w$ and constants $k, n$ so that the integral has the form $\int k w^{n} d w$.
$$\int \frac{\cos t}{\sin t} d t$$

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Problem 103

Find a substitution $w$ and constants $k, n$ so that the integral has the form $\int k w^{n} d w$.
$$\int \frac{2 x d x}{\left(x^{2}-3\right)^{2}}$$

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Problem 104

Find constants $k, n, w_{0}, w_{1}$ so the the integral has the form $\int_{w_{0}}^{w_{1}} k w^{n} d w$.
$$\int_{1}^{5} \frac{3 x d x}{\sqrt{5 x^{2}+7}}, \quad w=5 x^{2}+7$$

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Problem 105

Find constants $k, n, w_{0}, w_{1}$ so the the integral has the form $\int_{w_{0}}^{w_{1}} k w^{n} d w$.
$$\int_{0}^{5} \frac{2^{x} d x}{2^{x}+3}, \quad w=2^{x}+3$$

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Problem 106

Find constants $k, n, w_{0}, w_{1}$ so the the integral has the form $\int_{w_{0}}^{w_{1}} k w^{n} d w$.
$$\int_{\pi / 12}^{\pi / 4} \sin ^{7}(2 x) \cos (2 x) d x, \quad w=\sin 2 x$$

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Problem 107

Find a substitution $w$ and a constant $k$ so that the integral has the form $\int k e^{w} d w$.
$$\int x e^{-x^{2}} d x$$

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Problem 108

Find a substitution $w$ and a constant $k$ so that the integral has the form $\int k e^{w} d w$.
$$\int e^{\sin \phi} \cos \phi d \phi$$

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Problem 109

Find a substitution $w$ and a constant $k$ so that the integral has the form $\int k e^{w} d w$.
$$\int \sqrt{e^{r}} d r$$

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Problem 110

Find a substitution $w$ and a constant $k$ so that the integral has the form $\int k e^{w} d w$.
$$\int \frac{z^{2} d z}{e^{-z^{3}}}$$

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Problem 111

Find a substitution $w$ and a constant $k$ so that the integral has the form $\int k e^{w} d w$.
$$\int e^{2 t} e^{3 t-4} d t$$

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Problem 112

In Problems $112-113,$ find a substitution $w$ and constants $a, b, A$ so that the integral has the form $\int_{a}^{b} A e^{w} d w$
$$\int_{3}^{7} e^{2 t-3} d t$$

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Problem 113

Find a substitution $w$ and constants $a, b, A$ so that the integral has the form $\int_{a}^{b} A e^{w} d w$.
$$\int_{0}^{1} e^{\cos (\pi t)} \sin (\pi t) d t$$

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Problem 114

Integrate:
(a) $\int \frac{1}{\sqrt{x}} d x$
(b) $\int \frac{1}{\sqrt{x+1}} d x$
(c) $\int \frac{1}{\sqrt{x}+1} d x$

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Problem 115

If appropriate, evaluate the following integrals by substitution. If substitution is not appropriate, say so, and do not evaluate.
(a) $\int x \sin \left(x^{2}\right) d x$
(b) $\int x^{2} \sin x d x$
(c) $\int \frac{x^{2}}{1+x^{2}} d x$
(d) $\int \frac{x}{\left(1+x^{2}\right)^{2}} d x$
(e) $\int x^{3} e^{x^{2}} d x$
(f) $\int \frac{\sin x}{2+\cos x} d x$

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Problem 116

Find the exact area.
Under $f(x)=x e^{x^{2}}$ between $x=0$ and $x=2$.

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Problem 117

Find the exact area.
Under $f(x)=1 /(x+1)$ between $x=0$ and $x=2$.

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Problem 118

Find the exact area.
Under $f(x)=\sinh (x / 2)$ between $x=0$ and $x=2$.

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Problem 119

Find the exact area.
$$\text { Under } f(\theta)=\left(e^{\theta+1}\right)^{3} \text { for } 0 \leq \theta \leq 2$$

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Problem 120

Find the exact area.
Between $e^{t}$ and $e^{t+1}$ for $0 \leq t \leq 2$.

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Problem 121

Find the exact area.
Between $y=e^{x}, y=3,$ and the $y$ -axis.

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Problem 122

Find the exact area.
Under one arch of the curve $V(t)=V_{0} \sin (\omega t),$ where $V_{0}>0$ and $\omega>0$.

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Problem 123

Find the exact average value of $f(x)=1 /(x+1)$ on the interval $x=0$ to $x=2 .$ Sketch a graph showing the function and the average value.

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Problem 124

Let $g(x)=f(2 x) .$ Show that the average value of $f$ on the interval $[0,2 b]$ is the same as the average value of $g$ on the interval $[0, b]$.

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Problem 125

Suppose $\int_{0}^{2} g(t) d t=5 .$ Calculate the following:
(b) $\int_{0}^{2} g(2-t) d t$
(a) $\int_{0}^{4} g(t / 2) d t$

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Problem 126

Suppose $\int_{0}^{1} f(t) d t=3 .$ Calculate the following:
(a) $\int_{0}^{0.5} f(2 t) d t$
(b) $\int_{0}^{1} f(1-t) d t$
(c) $\int_{1}^{1.5} f(3-2 t) d t$

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Problem 127

(a) Calculate exactly: $\int_{-\pi}^{\pi} \cos ^{2} \theta \sin \theta d \theta$.
(b) Calculate the exact area under the curve $y=\cos ^{2} \theta \sin \theta$ between $\theta=0$ and $\theta=\pi$.

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Problem 128

Find $\int 4 x\left(x^{2}+1\right) d x$ using two methods:
(a) Do the multiplication first, and then antidifferentiate.
(b) Use the substitution $w=x^{2}+1$.
(c) Explain how the expressions from parts (a) and (b) are different. Are they both correct?

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Problem 129

(a) Find $\int \sin \theta \cos \theta d \theta$.
(b) You probably solved part (a) by making the substitution $w=\sin \theta$ or $w=\cos \theta .$ (If not, go back and do it that way.) Now find $\int \sin \theta \cos \theta d \theta$ by making the other substitution.
(c) There is yet another way of finding this integral which involves the trigonometric identities $\sin (2 \theta)=2 \sin \theta \cos \theta$ $\cos (2 \theta)=\cos ^{2} \theta-\sin ^{2} \theta$.
Find $\int \sin \theta \cos \theta d \theta$ using one of these identities and then the substitution $w=2 \theta$.
(d) You should now have three different expressions for the indefinite integral $\int \sin \theta \cos \theta d \theta .$ Are they really different? Are they all correct? Explain.

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Problem 130

Find a substitution $w$ and constants $a, b, k$ so that the integral has the form $\int_{a}^{b} k f(w) d w$.
$$\int_{1}^{9} f(6 x \sqrt{x}) \sqrt{x} d x$$

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Problem 131

Find a substitution $w$ and constants $a, b, k$ so that the integral has the form $\int_{a}^{b} k f(w) d w$.
$$\int_{2}^{5} \frac{f\left(\ln \left(x^{2}+1\right)\right) x d x}{x^{2}+1}$$

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Problem 132

Find the solution of the initial value problem
$$
y^{\prime}=\tan x+1, \quad y(0)=1
$$

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Problem 133

Let $I_{m, n}=\int_{0}^{1} x^{m}(1-x)^{n} d x$ for constant $m, n .$ Show that $I_{m, n}=I_{n, m}$.

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Problem 134

Let $f(t)$ be the velocity in meters/second of a car at time $t$ in seconds. Give an integral for the change of position of the car
(a) For the time interval $0 \leq t \leq 60$.
(b) In terms of $T$ in minutes, for the same time interval.

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Problem 135

Over the past fifty years the carbon dioxide level in the atmosphere has increased. Carbon dioxide is believed to drive temperature, so predictions of future carbon dioxide levels are important. If $C(t)$ is carbon dioxide level in parts per million (ppm) and $t$ is time in years since 1950 three possible models are: $^{1}$
I $C^{\prime}(t)=1.3$
II $C^{\prime}(t)=0.5+0.03 t$
III $C^{\prime}(t)=0.5 e^{0.02 t}$
(a) Given that the carbon dioxide level was 311 ppm in $1950,$ find $C(t)$ for each model.
(b) Find the carbon dioxide level in 2020 predicted by each model.

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Problem 136

Let $f(t)$ be the rate of flow, in cubic meters per hour, of a flooding river at time $t$ in hours. Give an integral for the total flow of the river
(a) Over the 3 -day period $0 \leq t \leq 72$.
(b) In terms of time $T$ in days over the same 3 -day period.

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Problem 137

With $t$ in years since $2000,$ the population, $P,$ of the world in billions can be modeled by $P=6.1 e^{0.012 t}$
(a) What does this model predict for the world population in $2010 ?$ In $2020 ?$
(b) Use the Fundamental Theorem to predict the average population of the world between 2000 and 2010

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Problem 138

Oil is leaking out of a ruptured tanker at the rate of $r(t)=50 e^{-0.02 t}$ thousand liters per minute.
(a) At what rate, in liters per minute, is oil leaking out at $t=0 ?$ At $t=60 ?$
(b) How many liters leak out during the first hour?

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Problem 139

Throughout much of the $20^{\text {th }}$ century, the yearly consumption of electricity in the US increased exponentially at a continuous rate of $7 \%$ per year. Assume this trend continues and that the electrical energy consumed in 1900 was 1.4 million megawatt-hours.
(a) Write an expression for yearly electricity consumption as a function of time, $t,$ in years since 1900
(b) Find the average yearly electrical consumption throughout the $20^{\text {th }}$ century.
(c) During what year was electrical consumption closest to the average for the century?
(d) Without doing the calculation for part (c), how could you have predicted which half of the century the answer would be in?

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Problem 140

An electric current, $I(t),$ flowing out of a capacitor, decays according to $I(t)=I_{0} e^{-t},$ where $t$ is time. Find the charge, $Q(t),$ remaining in the capacitor at time $t$ The initial charge is $Q_{0}$ and $Q(t)$ is related to $I(t)$ by
$$
Q^{\prime}(t)=-I(t)
$$

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Problem 141

If we assume that wind resistance is proportional to velocity, then the downward velocity, $v,$ of a body of mass $m$ falling vertically is given by
$$
v=\frac{m g}{k}\left(1-e^{-k t / m}\right)
$$,
where $g$ is the acceleration due to gravity and $k$ is a constant. Find the height, $h$, above the surface of the earth as a function of time. Assume the body starts at height $h_{0}$.

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Problem 142

If we assume that wind resistance is proportional to the square of velocity, then the downward velocity, $v,$ of a falling body is given by
$$
v=\sqrt{\frac{g}{k}}\left(\frac{e^{t \sqrt{g k}}-e^{-t \sqrt{g k}}}{e^{t \sqrt{g k}}+e^{-t \sqrt{g k}}}\right)
$$
Use the substitution $w=e^{t \sqrt{g k}}+e^{-t \sqrt{g k}}$ to find the height, $h,$ of the body above the surface of the earth as a function of time. Assume the body starts at a height $h_{0}$.

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Problem 143

(a) Between 2000 and 2010 , ACME Widgets sold widgets at a continuous rate of $R=R_{0} e^{0.125 t}$ widgets per year, where $t$ is time in years since January 1 2000. Suppose they were selling widgets at a rate of 1000 per year on January $1,2000 .$ How many widgets did they sell between 2000 and $2010 ?$ How many did they sell if the rate on January 1,2000 was 1,000,000 widgets per year?
(b) In the first case ( 1000 widgets per year on January 1, 2000 ), how long did it take for half the widgets in the ten-year period to be sold? In the second case $(1,000,000 \text { widgets per year on January } 1,2000)$ when had half the widgets in the ten-year period been sold?
(c) In $2010,$ ACME advertised that half the widgets it had sold in the previous ten years were still in use. Based on your answer to part (b), how long must a widget last in order to justify this claim?

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Problem 144

The rate at which water is flowing into a tank is $r(t)$ gallons/minute, with $t$ in minutes.
(a) Write an expression approximating the amount of water entering the tank during the interval from time $t$ to time $t+\Delta t,$ where $\Delta t$ is small.
(b) Write a Riemann sum approximating the total amount of water entering the tank between $t=0$ and $t=5 .$ Write an exact expression for this amount.
(c) By how much has the amount of water in the tank changed between $t=0$ and $t=5$ if $r(t)=$ $20 e^{0.02 t} ?$
(d) If $r(t)$ is as in part (c), and if the tank contains 3000 gallons initially, find a formula for $Q(t),$ the amount of water in the tank at time $t$

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Problem 145

Explain what is wrong with the statement.
$$\int(f(x))^{2} d x=(f(x))^{3} / 3+C$$

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Problem 146

Explain what is wrong with the statement.
$$\int \cos \left(x^{2}\right) d x=\sin \left(x^{2}\right) /(2 x)+C$$

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Problem 147

Explain what is wrong with the statement.
$$\int_{0}^{\pi / 2} \cos (3 x) d x=(1 / 3) \int_{0}^{\pi / 2} \cos w d w$$

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Problem 148

Give an example of:
A possible $f(\theta)$ so that the following integral can be integrated by substitution:
$$
\int f(\theta) e^{\cos \theta} d \theta
$$

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Problem 149

Give an example of:
An indefinite integral involving $\sin \left(x^{3}-3 x\right)$ that can be evaluated by substitution.

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Problem 150

Decide whether the statements are true or false. Give an explanation for your answer.
$$\int f^{\prime}(x) \cos (f(x)) d x=\sin (f(x))+C$$

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Problem 151

Decide whether the statements are true or false. Give an explanation for your answer.
$$\int(1 / f(x)) d x=\ln |f(x)|+C$$

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Problem 152

Decide whether the statements are true or false. Give an explanation for your answer.
$\int t \sin \left(5-t^{2}\right) d t$ can be evaluated using substitution.

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