## Problem Set: Trigonometric Integrals

Fill in the blank to make a true statement.

1. ${\sin}^{2}x+\_\_\_\_\_\_\_=1$

2. ${\sec}^{2}x - 1=\_\_\_\_\_\_\_$

Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power.

3. ${\sin}^{2}x=\_\_\_\_\_\_\_$

4. ${\cos}^{2}x=\_\_\_\_\_\_\_$

Evaluate each of the following integrals by u-substitution.

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

6. $\displaystyle\int \sqrt{\cos{x}}\sin{x}dx$

7. $\displaystyle\int {\tan}^{5}\left(2x\right){\sec}^{2}\left(2x\right)dx$

8. $\displaystyle\int {\sin}^{7}\left(2x\right)\cos\left(2x\right)dx$

9. $\displaystyle\int \tan\left(\frac{x}{2}\right){\sec}^{2}\left(\frac{x}{2}\right)dx$

10. $\displaystyle\int {\tan}^{2}x{\sec}^{2}xdx$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

11. $\displaystyle\int {\sin}^{3}xdx$

12. $\displaystyle\int {\cos}^{3}xdx$

13. $\displaystyle\int \sin{x}\cos{x}dx$

14. $\displaystyle\int {\cos}^{5}xdx$

15. $\displaystyle\int {\sin}^{5}x{\cos}^{2}xdx$

16. $\displaystyle\int {\sin}^{3}x{\cos}^{3}xdx$

17. $\displaystyle\int \sqrt{\sin{x}}\cos{x}dx$

18. $\displaystyle\int \sqrt{\sin{x}}{\cos}^{3}xdx$

19. $\displaystyle\int \sec{x}\tan{x}dx$

20. $\displaystyle\int \tan\left(5x\right)dx$

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

22. $\displaystyle\int \tan{x}{\sec}^{3}xdx$

23. $\displaystyle\int {\sec}^{4}xdx$

24. $\displaystyle\int \cot{x}dx$

25. $\displaystyle\int \csc{x}dx$

26. $\displaystyle\int \frac{{\tan}^{3}x}{\sqrt{\sec{x}}}dx$

For the following exercises, find a general formula for the integrals.

27. $\displaystyle\int {\sin}^{2}ax\cos{ax} dx$

28. $\displaystyle\int \sin{ax} \cos{ax} dx$.

Use the double-angle formulas to evaluate the following integrals.

29. ${\displaystyle\int }_{0}^{\pi }{\sin}^{2}xdx$

30. ${\displaystyle\int }_{0}^{\pi }{\sin}^{4}xdx$

31. $\displaystyle\int {\cos}^{2}3xdx$

32. $\displaystyle\int {\sin}^{2}x{\cos}^{2}xdx$

33. $\displaystyle\int {\sin}^{2}xdx+\displaystyle\int {\cos}^{2}xdx$

34. $\displaystyle\int {\sin}^{2}x{\cos}^{2}\left(2x\right)dx$

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.

35. ${\displaystyle\int }_{0}^{2\pi }\cos{x}\sin2xdx$

36. ${\displaystyle\int }_{0}^{\pi }\sin3x\sin5xdx$

37. ${\displaystyle\int }_{0}^{\pi }\cos\left(99x\right)\sin\left(101x\right)dx$

38. ${\displaystyle\int }_{\text{-}\pi }^{\pi }{\cos}^{2}\left(3x\right)dx$

39. ${\displaystyle\int }_{0}^{2\pi }\sin{x}\sin\left(2x\right)\sin\left(3x\right)dx$

40. ${\displaystyle\int }_{0}^{4\pi }\cos\left(\frac{x}{2}\right)\sin\left(\frac{x}{2}\right)dx$

41. ${\displaystyle\int }_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{{\cos}^{3}x}{\sqrt{\sin{x}}}dx$ (Round this answer to three decimal places.)

42. ${\displaystyle\int }_{\frac{\text{-}\pi}{3}}^{\frac{\pi}{3}}\sqrt{{\sec}^{2}x - 1}dx$

43. ${\displaystyle\int }_{0}^{\frac{\pi}{2}}\sqrt{1-\cos\left(2x\right)}dx$

44. Find the area of the region bounded by the graphs of the equations $y=\sin{x},y={\sin}^{3}x,x=0,\text{and }x=\frac{\pi }{2}$.

45. Find the area of the region bounded by the graphs of the equations $y={\cos}^{2}x,y={\sin}^{2}x,x=-\frac{\pi }{4},\text{and }x=\frac{\pi }{4}$.

46. A particle moves in a straight line with the velocity function $v\left(t\right)=\sin\left(\omega t\right){\cos}^{2}\left(\omega t\right)$. Find its position function $x=f\left(t\right)$ if $f\left(0\right)=0$.

47. Find the average value of the function $f\left(x\right)={\sin}^{2}x{\cos}^{3}x$ over the interval $\left[\text{-}\pi ,\pi \right]$.

For the following exercises, solve the differential equations.

48. $\frac{dy}{dx}={\sin}^{2}x$. The curve passes through point $\left(0,0\right)$.

49. $\frac{dy}{d\theta }={\sin}^{4}\left(\pi \theta \right)$

50. Find the length of the curve $y=\text{ln}\left(\csc{x}\right),\frac{\pi }{4}\le x\le \frac{\pi }{2}$.

51. Find the length of the curve $y=\text{ln}\left(\sin{x}\right),\frac{\pi }{3}\le x\le \frac{\pi }{2}$.

52. Find the volume generated by revolving the curve $y=\cos\left(3x\right)$ about the x-axis, $0\le x\le \frac{\pi }{36}$.

For the following exercises, use this information: The inner product of two functions f and g over $\left[a,b\right]$ is defined by $f\left(x\right)\cdot g\left(x\right)=\langle f,g\rangle ={\displaystyle\int }_{a}^{b}f\cdot gdx$. Two distinct functions f and g are said to be orthogonal if $\langle f,g\rangle =0$.

53. Show that $\left\{\sin\left(2x\right),\cos\left(3x\right)\right\}$ are orthogonal over the interval $\left[\text{-}\pi ,\pi \right]$.

54. Evaluate ${\displaystyle\int }_{\text{-}\pi }^{\pi }\sin\left(mx\right)\cos\left(nx\right)dx$.

55. Integrate ${y}^{\prime }=\sqrt{\tan{x}}{\sec}^{4}x$.

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning.

56. $\displaystyle\int {\sin}^{456}x\cos{x}dx$ or $\displaystyle\int {\sin}^{2}x{\cos}^{2}xdx$

57. $\displaystyle\int {\tan}^{350}x{\sec}^{2}xdx$ or $\displaystyle\int {\tan}^{350}x\sec{x}dx$