Thomas Calculus

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Chapter 8

Techniques Of Integration - all with Video Answers

Educators

Section 1

Basic Integration Formulas

Problem 1

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{16 x d x}{\sqrt{8 x^{2}+1}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 2

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{3 \cos x d x}{\sqrt{1+3 \sin x}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 3

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int 3 \sqrt{\sin v} \cos v d v
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 4

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \cot ^{3} y \csc ^{2} y d y
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 5

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int_{0}^{1} \frac{16 x d x}{8 x^{2}+2}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 6

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int_{\pi / 4}^{\pi / 3} \frac{\sec ^{2} z}{\tan z} d z
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 7

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{d x}{\sqrt{x}(\sqrt{x}+1)}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 8

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{d x}{x-\sqrt{x}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 9

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \cot (3-7 x) d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 10

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \csc (\pi x-1) d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 11

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int e^{\theta} \csc \left(e^{\theta}+1\right) d \theta
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 12

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{\cot (3+\ln x)}{x} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 13

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \sec \frac{t}{3} d t
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 14

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int x \sec \left(x^{2}-5\right) d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 15

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \csc (s-\pi) d s
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 16

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{1}{\theta^{2}} \csc \frac{1}{\theta} d \theta
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 17

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int_{0}^{\sqrt{\ln 2}} 2 x e^{x^{2}} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 18

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int_{\pi / 2}^{\pi}(\sin y) e^{\cos y} d y
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 19

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int e^{\tan v} \sec ^{2} v d v
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 20

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{e^{\sqrt{t}} d t}{\sqrt{t}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 21

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int 3^{x+1} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 22

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{2^{\ln x}}{x} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 23

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{2^{\sqrt{w}} d w}{2 \sqrt{w}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 24

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int 10^{2 \theta} d \theta
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 25

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{9 d u}{1+9 u^{2}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 26

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{4 d x}{1+(2 x+1)^{2}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 27

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int_{0}^{1 / 6} \frac{d x}{\sqrt{1-9 x^{2}}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 28

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int_{0}^{1} \frac{d t}{\sqrt{4-t^{2}}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 29

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{2 s d s}{\sqrt{1-s^{4}}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 30

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{2 d x}{x \sqrt{1-4 \ln ^{2} x}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 31

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{6 d x}{x \sqrt{25 x^{2}-1}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 32

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{d r}{r \sqrt{r^{2}-9}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 33

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{d x}{e^{x}+e^{-x}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 34

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{d y}{\sqrt{e^{2 y}-1}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 35

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int_{1}^{e^{\pi / 3}} \frac{d x}{x \cos (\ln x)}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 36

Evaluate each integral in Exercises $1-36$ by using a substitution to reduce it to standard form.
$$
\int \frac{\ln x d x}{x+4 x \ln ^{2} x}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 37

Evaluate each integral in Exercises $37-42$ by completing the square and using a substitution to reduce it to standard form.
$$
\int_{1}^{2} \frac{8 d x}{x^{2}-2 x+2}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 38

Evaluate each integral in Exercises $37-42$ by completing the square and using a substitution to reduce it to standard form.
$$
\int_{2}^{4} \frac{2 d x}{x^{2}-6 x+10}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 39

Evaluate each integral in Exercises $37-42$ by completing the square and using a substitution to reduce it to standard form.
$$
\int \frac{d t}{\sqrt{-t^{2}+4 t-3}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 40

Evaluate each integral in Exercises $37-42$ by completing the square and using a substitution to reduce it to standard form.
$$
\int \frac{d \theta}{\sqrt{2 \theta-\theta^{2}}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 41

Evaluate each integral in Exercises $37-42$ by completing the square and using a substitution to reduce it to standard form.
$$
\int \frac{d x}{(x+1) \sqrt{x^{2}+2 x}} \quad 42
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 42

Evaluate each integral in Exercises $37-42$ by completing the square and using a substitution to reduce it to standard form.
$$
\int \frac{d x}{(x-2) \sqrt{x^{2}-4 x+3}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 43

Evaluate each integral in Exercises $43-46$ by using trigonometric identities and substitutions to reduce it to standard form.
$$
\int(\sec x+\cot x)^{2} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 44

Evaluate each integral in Exercises $43-46$ by using trigonometric identities and substitutions to reduce it to standard form.
$$
\int(\csc x-\tan x)^{2} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 45

Evaluate each integral in Exercises $43-46$ by using trigonometric identities and substitutions to reduce it to standard form.
$$
\int \csc x \sin 3 x d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 46

Evaluate each integral in Exercises $43-46$ by using trigonometric identities and substitutions to reduce it to standard form.
$$
\int(\sin 3 x \cos 2 x-\cos 3 x \sin 2 x) d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 47

Evaluate each integral in Exercises $47-52$ by reducing the improper fraction and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{x}{x+1} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 48

Evaluate each integral in Exercises $47-52$ by reducing the improper fraction and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{x^{2}}{x^{2}+1} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 49

Evaluate each integral in Exercises $47-52$ by reducing the improper fraction and using a substitution (if necessary) to reduce it to standard form.
$$
\int_{\sqrt{2}}^{3} \frac{2 x^{3}}{x^{2}-1} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 50

Evaluate each integral in Exercises $47-52$ by reducing the improper fraction and using a substitution (if necessary) to reduce it to standard form.
$$
\int_{-1}^{3} \frac{4 x^{2}-7}{2 x+3} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 51

Evaluate each integral in Exercises $47-52$ by reducing the improper fraction and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{4 t^{3}-t^{2}+16 t}{t^{2}+4} d t
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 52

Evaluate each integral in Exercises $47-52$ by reducing the improper fraction and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{2 \theta^{3}-7 \theta^{2}+7 \theta}{2 \theta-5} d \theta
$$

Khushbu Rani

Numerade Educator

Problem 53

Evaluate each integral in Exercises $53-56$ by separating the fraction and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{1-x}{\sqrt{1-x^{2}}} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 54

Evaluate each integral in Exercises $53-56$ by separating the fraction and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{x+2 \sqrt{x-1}}{2 x \sqrt{x-1}} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 55

Evaluate each integral in Exercises $53-56$ by separating the fraction and using a substitution (if necessary) to reduce it to standard form.
$$
\int_{0}^{\pi / 4} \frac{1+\sin x}{\cos ^{2} x} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 56

Evaluate each integral in Exercises $53-56$ by separating the fraction and using a substitution (if necessary) to reduce it to standard form.
$$
\int_{0}^{1 / 2} \frac{2-8 x}{1+4 x^{2}} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 57

Evaluate each integral in Exercises $57-62$ by multiplying by a form of 1 and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{1}{1+\sin x} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 58

Evaluate each integral in Exercises $57-62$ by multiplying by a form of 1 and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{1}{1+\cos x} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 59

Evaluate each integral in Exercises $57-62$ by multiplying by a form of 1 and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{1}{\sec \theta+\tan \theta} d \theta
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 60

Evaluate each integral in Exercises $57-62$ by multiplying by a form of 1 and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{1}{\csc \theta+\cot \theta} d \theta
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 61

Evaluate each integral in Exercises $57-62$ by multiplying by a form of 1 and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{1}{1-\sec x} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 62

Evaluate each integral in Exercises $57-62$ by multiplying by a form of 1 and using a substitution (if necessary) to reduce it to standard form.
$$
\int \frac{1}{1-\csc x} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 63

Evaluate each integral in Exercises $63-70$ by eliminating the square root.
$$
\int_{0}^{2 \pi} \sqrt{\frac{1-\cos x}{2}} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 64

Evaluate each integral in Exercises $63-70$ by eliminating the square root.
$$
\int_{0}^{\pi} \sqrt{1-\cos 2 x} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 65

Evaluate each integral in Exercises $63-70$ by eliminating the square root.
$$
\int_{\pi / 2}^{\pi} \sqrt{1+\cos 2 t} d t
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 66

Evaluate each integral in Exercises $63-70$ by eliminating the square root.
$$
\int_{-\pi}^{0} \sqrt{1+\cos t} d t
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 67

Evaluate each integral in Exercises $63-70$ by eliminating the square root.
$$
\int_{-\pi}^{0} \sqrt{1-\cos ^{2} \theta} d \theta
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 68

Evaluate each integral in Exercises $63-70$ by eliminating the square root.
$$
\int_{\pi / 2}^{\pi} \sqrt{1-\sin ^{2} \theta} d \theta
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 69

Evaluate each integral in Exercises $63-70$ by eliminating the square root.
$$
\int_{-\pi / 4}^{\pi / 4} \sqrt{1+\tan ^{2} y} d y
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 70

Evaluate each integral in Exercises $63-70$ by eliminating the square root.
$$
\int_{\pi / 4}^{0} \sqrt{\sec ^{2} y-1} d y
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 71

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int_{\pi / 4}^{3 \pi / 4}(\csc x-\cot x)^{2} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 72

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int_{0}^{\pi / 4}(\sec x+4 \cos x)^{2} d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 73

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int \cos \theta \csc (\sin \theta) d \theta
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 74

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int\left(1+\frac{1}{x}\right) \cot (x+\ln x) d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 75

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int(\csc x-\sec x)(\sin x+\cos x) d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 76

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int 3 \sinh \left(\frac{x}{2}+\ln 5\right) d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 77

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int 3 \sinh \left(\frac{x}{2}+\ln 5\right) d x
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 78

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int \frac{d x}{x \sqrt{4 x^{2}-1}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 79

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int \frac{7 d x}{(x-1) \sqrt{x^{2}-2 x-48}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 80

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int \frac{d x}{(2 x+1) \sqrt{4 x^{2}+4 x}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 81

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int \sec ^{2} t \tan (\tan t) d t
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 82

Evaluate each integral in Exercises $71-82$ by using any technique you think is appropriate.
$$
\int \frac{d x}{x \sqrt{3+x^{2}}}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 83

a. Evaluate $\int \cos ^{3} \theta d \theta$ . (Hint: $\cos ^{2} \theta=1-\sin ^{2} \theta . )$
b. Evaluate $\int \cos ^{5} \theta d \theta$
c. Without actually evaluating the integral, explain how you would evaluate $\int \cos ^{9} \theta d \theta$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 84

a. Evaluate $\int \sin ^{3} \theta d \theta$ . (Hint: $\sin ^{2} \theta=1-\cos ^{2} \theta . )$
b. Evaluate $\int \sin ^{5} \theta d \theta$
c. Evaluate $\int \sin ^{7} \theta d \theta$
d. Without actually evaluating the integral, explain how you would evaluate $\int \sin ^{13} \theta d \theta$

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 85

a. Express $\int \tan ^{3} \theta d \theta$ in terms of $\int \tan \theta d \theta$ . Then evaluate
$\int \tan ^{3} \theta d \theta \cdot\left(\text {Hint} : \tan ^{2} \theta=\sec ^{2} \theta-1 .\right)$
b. Express $\int \tan ^{5} \theta d \theta$ in terms of $\int \tan ^{3} \theta d \theta$
c. Express $\int \tan ^{7} \theta d \theta$ in terms of $\int \tan ^{5} \theta d \theta$
d. Express $\int \tan ^{2 k+1} \theta d \theta,$ where $k$ is a positive integer, in terms of $\int \tan ^{2 k-1} \theta d \theta$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 86

a. Express $\int \cot ^{3} \theta d \theta$ in terms of $\int \cot \theta d \theta .$ Then evaluate
$\int \cot ^{3} \theta d \theta .\left(\text {Hint} : \cot ^{2} \theta=\csc ^{2} \theta-1 .\right)$
b. Express $\int \cot ^{5} \theta d \theta$ in terms of $\int \cot ^{3} \theta d \theta$
c. Express $\int \cot ^{7} \theta d \theta$ in terms of $\int \cot ^{5} \theta d \theta$
d. Express $\int \cot ^{2 k+1} \theta d \theta,$ where $k$ is a positive integer, in terms of $\int \cot ^{2 k-1} \theta d \theta$

Angela Guo

Numerade Educator

Problem 87

Area Find the area of the region bounded above by $y=2 \cos x$ and below by $y=\sec x,-\pi / 4 \leq x \leq \pi / 4$

Prakash Hampole

Prakash Hampole

Numerade Educator

Problem 88

Area Find the area of the "triangular" region that is bounded from above and below by the curves $y=\csc x$ and $y=\sin x,$ $\pi / 6 \leq x \leq \pi / 2,$ and on the left by the line $x=\pi / 6$ .

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 89

Volume Find the volume of the solid generated by revolving the region in Exercise 87 about the $x$ -axis.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 90

Volume Find the volume of the solid generated by revolving the region in Exercise 88 about the $x$ raxis

Carolyn Behr-Jerome

Carolyn Behr-Jerome

Numerade Educator

Problem 91

Are length Find the length of the curve $y=\ln (\cos x)$ $0 \leq x \leq \pi / 3$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 92

Are length Find the length of the curve $y=\ln (\sec x)$ $0 \leq x \leq \pi / 4$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 93

Centroid Find the centroid of the region bounded by the $x$ -axis, the curve $y=\sec x,$ and the lines $x=-\pi / 4, x=\pi / 4$ .

Doruk Isik

Numerade Educator

Problem 94

Centroid Find the centroid of the region that is bounded by the $x$ -axis, the curve $y=\csc x,$ and the lines $x=\pi / 6, x=5 \pi / 6$

Matthew Allcock

Matthew Allcock

Numerade Educator

Problem 95

The integral of cse $x$ Repeat the derivation in Example $7,$ using cofunctions, to show that

Lucas Finney

Numerade Educator

Problem 96

Using different substitutions Show that the integral
$$
\int\left(\left(x^{2}-1\right)(x+1)\right)^{-2 / 3} d x
$$
can be evaluated with any of the following substitutions.
a. $u=1 /(x+1)$
b. $u=((x-1) /(x+1))^{k}$ for $k=1,1 / 2,1 / 3,-1 / 3,-2 / 3$
c. $u=\tan ^{-1} x$
d. $u=\tan ^{-1} \sqrt{x}$
e. $u=\tan ^{-1}((x-1) / 2)$
f. $u=\cos ^{-1} x$
g. $u=\cosh ^{-1} x$
What is the value of the integral? (Source: "Problems and Solutions," College Mathematics Journal, Vol. $21,$ No. 5 (Nov. 1990$)$ , pp. $425-426 .$ )

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator