1. True or False To find $\int \cos ^{5}x\,dx,$ factor out $\cos x$ and use the identity $\cos ^{2}x=1-\sin ^{2}x.$

2. True or False To find $\int \sin (2x) \cos (3x) \,dx,$ use a product-to-sum identity.

3. $\int \cos ^{5}x\,dx$

4. $\int \sin ^{3}x\,dx$

5. $\int \sin ^{6}x\,dx$

6. $\int \cos ^{4}x\,dx$

7. $\int \sin ^{2}\left( \pi x\right) \,dx$

8. $\int \cos ^{4}(2x) \,dx$

9. $\int_{0}^{\pi }\cos ^{5}x~dx$

10. $\int_{-\pi /3}^{\pi /3}\sin ^{3}x\,dx$

11. $\int \sin ^{3}x\cos ^{2}x\,dx$

12. $\int \sin ^{4}x\cos ^{3}x\,dx$

13. $\int \sin ^{2}x\cos ^{2}x\,dx$

14. $\int \sin ^{4}x\cos ^{2}x\,dx$

15. $\int \sin x\cos ^{1/3}x\,dx$

16. $\int \cos ^{3}x\sin ^{1/2}x\,dx$

17. $\int \sin ^{2}\left(\dfrac{x}{2}\right) \cos ^{3}\left( \dfrac{x}{2}\right) \,dx$

18. $\int \sin ^{3}(4x) \cos ^{3}(4x) \,dx$

19. $\int \tan ^{3}x\sec ^{2}x\,dx$

20. $\int \tan x\sec ^{5}x\,dx$

21. $\int \tan ^{2}x\sec ^{2}x\,dx$

22. $\int \tan ^{5}x\sec ^{2}x\,dx$

23. $\int \tan ^{2}x\sec ^{3}x\,dx$

24. $\int \tan ^{4}x\,\sec x\,dx$

25. $\int \cot ^{3}x\csc x\,dx$

26. $\int \cot ^{3}x\csc ^{2}x\,dx$

27. $\int \sin (3x) \cos x\,dx$

28. $\int \sin x\cos (3x) \,dx$

29. $\int \cos x\cos (3x) \,dx$

30. $\int \cos (2x) \cos x\,dx$

31. $\int \sin (2x) \sin (4x) \,dx$

32. $\int \sin (3x) \sin x\, dx$

33. $\int_{0}^{\pi /2}\sin (2x) \sin x\,dx$

34. $\int_{0}^{\pi }\cos x\cos (4x) \,dx$

35. $\int \sin ^{2}x\cos x\,dx$

36. $\int \sin ^{3}x\cos x\,dx$

37. $\int \dfrac{\sin x\,dx}{\cos ^{2}x}$

38. $\int \dfrac{\cos x\,dx}{\sin ^{4}x}$

39. $\int \cos ^{3}(3x) \,dx$

40. $\int \sin ^{5}(3x) \,dx$

41. $\int_{0}^{\pi }\sin ^{3}x\cos ^{5}x\,dx$

42. $\int_{0}^{\pi /2}\sin ^{3}x\cos ^{3}x\,dx$

43. $\int \tan ^{3}x\,dx$

44. $\int \cot ^{5}x\,dx$

45. $\int \dfrac{\sec ^{6}x}{\tan ^{3}x}\,dx$

46. $\int \tan ^{1/2}x\sec ^{2}x\,dx$

47. $\int \csc ^{2}x\cot ^{5}x\,dx$

48. $\int \cot x\csc ^{2}x\,dx$

487

49. $\int \cot(2x) \csc^{4}(2x)~dx$

50. $\int \cot^{2}(2x) \csc^{3}(2x)dx$

51. $\int_{0}^{\pi /4}\tan^{4}x\sec^{3}xdx$

52. $\int_{0}^{\pi /4}\tan^{2}x\sec xdx$

53. $\int \sin \left( \dfrac{x}{2}\right) \cos \left( \dfrac{3x}{2}\right)dx$

54. $\int \cos (-x) \sin (4x)dx$

55. $\int \sin \left( \dfrac{x}{2}\right) \sin \left( \dfrac{3x}{2}\right)dx$

56. $\int \cos (\pi x) \cos ( 3\pi x)dx$

57. Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of $y=\sin x$ and the $x$-axis from $x=0$ to $x=\pi$ about the $x$-axis. See the figure below.

    

58. Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of $y=\cos x$, $y=\sin x$, and $x=0$ from $x=0$ to $x=\dfrac{\pi }{4}$ about the $x$-axis.

    

59. Average Value

    1. (a) Find the average value of $f( x) =\sin x\cos ^{4}x$ over the interval $[ 0,\,\pi ]$.
    2. (b) Give a geometric interpretation to the average value.
    3. (c) Use graphing technology to graph $f$ and the average value on the same screen.

60. Rectilinear Motion The acceleration $a$ of an object at time $t$ is given by $a(t) =\cos ^{2}t\sin t$ m$/$s$^{2}$. At $t=0$, the object is at the origin and its speed is 5m$/$s. Find its distance from the origin at any time $t$.

61. Area and Volume Let $A$ be the area of the region in the first quadrant bounded by the graphs of $y=\sec x$, $y=2\sin x$, and the $y$-axis.

    1. (a) Find $A$.
    2. (b) Find the volume of the solid of revolution generated by revolving the region about the $x$-axis.

62. 

    1. (a) Use technology to graph the function $f( x) =\sin^{n}x,$ $0 \le x \le \pi$, for $n=5$, $n=10$, $n=20$, and $n=50$.
    2. (b) Find $\int_{0}^{\pi }\sin ^{n}xdx$ correct to three decimal places for $n=5$, $n=10$, $n=20$, and $n=50$.
    3. (c) What does (a) suggest about the shape of the graph of $f( x) =\sin ^{n}x$, $0 \le x \le \pi$, as $n\rightarrow \infty$.
    4. (d) Find $\lim\limits_{n\rightarrow \infty }\int_{0}^{\pi }\sin ^{n}x~dx$.

63. Find $\int \sin ^{4}x\,dx$.

    1. (a) Using the methods of this section.
    2. (b) Using the reduction formula given in Problem 61 in Section 7.1.
    3. (c) Verify that both results are equivalent.
    4. (d) Use a CAS to find $\int \sin ^{4}x~dx$.

64. 1. (a) Use the substitution $u=\sin x$ to find a function $f$ for which $$\int \sin x\cos x\,dx=f(x)+C_{1}.$$
    2. (b) Use the substitution $u=\cos x$ to find a function $g$ for which $$\int \sin x\cos x\,dx=g(x)+C_{2}.$$
    3. (c) Use the trigonometric identity $\sin (2x)=2\sin x\cos x$ to find a function $h$ for which $$\int \sin x\cos x\,dx=h(x)+C_{3}.$$
    4. (d) Compare the functions $f$ and $g$. Find a relationship between $C_{1}$ and $C_{2}$.
    5. (e) Compare the functions $f$ and $h$. Find a relationship between $C_{1}$ and $C_{3}$.

65. Derive a formula for $\int \sin (mx) \sin (nx) \,dx,\quad m ≠ n$.

66. Derive a formula for $\int \sin (mx) \cos (nx) \,dx,\quad m ≠ n$.

67. Derive a formula for $\int \cos (mx) \cos (nx) \,dx,\quad m ≠ n$.

68. Use the substitution $\sqrt{x}\,=\,\sin y$ to find $\int_{0}^{1/2}\dfrac{\sqrt{x}}{\sqrt{1-x}}\,dx$. $\bigg($Hint: sin$^{2}y\,{=}\,\dfrac{1\,{-}\,\cos (2y)}{2}.\bigg)$

69. Use an appropriate substitution to show that $$\int_{0}^{\pi /2}\sin ^{n}\theta \,d\theta =\,\int_{0}^{\pi /2}\cos ^{n}\theta \,d\theta .$$

70. 1. (a) What is wrong with the following?
       
       
    2. (b) Find $\int_{0}^{\pi }\cos ^{4}x~dx$.