Problem Set: Trigonometric Integrals

Fill in the blank to make a true statement.

1. [latex]{\sin}^{2}x+\_\_\_\_\_\_\_=1[/latex]

2. [latex]{\sec}^{2}x - 1=\_\_\_\_\_\_\_[/latex]

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

3. [latex]{\sin}^{2}x=\_\_\_\_\_\_\_[/latex]

4. [latex]{\cos}^{2}x=\_\_\_\_\_\_\_[/latex]

Evaluate each of the following integrals by u-substitution.

5. [latex]\displaystyle\int {\sin}^{3}x\cos{x}dx[/latex]

6. [latex]\displaystyle\int \sqrt{\cos{x}}\sin{x}dx[/latex]

7. [latex]\displaystyle\int {\tan}^{5}\left(2x\right){\sec}^{2}\left(2x\right)dx[/latex]

8. [latex]\displaystyle\int {\sin}^{7}\left(2x\right)\cos\left(2x\right)dx[/latex]

9. [latex]\displaystyle\int \tan\left(\frac{x}{2}\right){\sec}^{2}\left(\frac{x}{2}\right)dx[/latex]

10. [latex]\displaystyle\int {\tan}^{2}x{\sec}^{2}xdx[/latex]

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. [latex]\displaystyle\int {\sin}^{3}xdx[/latex]

12. [latex]\displaystyle\int {\cos}^{3}xdx[/latex]

13. [latex]\displaystyle\int \sin{x}\cos{x}dx[/latex]

14. [latex]\displaystyle\int {\cos}^{5}xdx[/latex]

15. [latex]\displaystyle\int {\sin}^{5}x{\cos}^{2}xdx[/latex]

16. [latex]\displaystyle\int {\sin}^{3}x{\cos}^{3}xdx[/latex]

17. [latex]\displaystyle\int \sqrt{\sin{x}}\cos{x}dx[/latex]

18. [latex]\displaystyle\int \sqrt{\sin{x}}{\cos}^{3}xdx[/latex]

19. [latex]\displaystyle\int \sec{x}\tan{x}dx[/latex]

20. [latex]\displaystyle\int \tan\left(5x\right)dx[/latex]

21. [latex]\displaystyle\int {\tan}^{2}x\sec{x}dx[/latex]

22. [latex]\displaystyle\int \tan{x}{\sec}^{3}xdx[/latex]

23. [latex]\displaystyle\int {\sec}^{4}xdx[/latex]

24. [latex]\displaystyle\int \cot{x}dx[/latex]

25. [latex]\displaystyle\int \csc{x}dx[/latex]

26. [latex]\displaystyle\int \frac{{\tan}^{3}x}{\sqrt{\sec{x}}}dx[/latex]

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

27. [latex]\displaystyle\int {\sin}^{2}ax\cos{ax} dx[/latex]

28. [latex]\displaystyle\int \sin{ax} \cos{ax} dx[/latex].

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

29. [latex]{\displaystyle\int }_{0}^{\pi }{\sin}^{2}xdx[/latex]

30. [latex]{\displaystyle\int }_{0}^{\pi }{\sin}^{4}xdx[/latex]

31. [latex]\displaystyle\int {\cos}^{2}3xdx[/latex]

32. [latex]\displaystyle\int {\sin}^{2}x{\cos}^{2}xdx[/latex]

33. [latex]\displaystyle\int {\sin}^{2}xdx+\displaystyle\int {\cos}^{2}xdx[/latex]

34. [latex]\displaystyle\int {\sin}^{2}x{\cos}^{2}\left(2x\right)dx[/latex]

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

35. [latex]{\displaystyle\int }_{0}^{2\pi }\cos{x}\sin2xdx[/latex]

36. [latex]{\displaystyle\int }_{0}^{\pi }\sin3x\sin5xdx[/latex]

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

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

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

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

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

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

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

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

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

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

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

For the following exercises, solve the differential equations.

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

49. [latex]\frac{dy}{d\theta }={\sin}^{4}\left(\pi \theta \right)[/latex]

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

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

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

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

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

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

55. Integrate [latex]{y}^{\prime }=\sqrt{\tan{x}}{\sec}^{4}x[/latex].

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

56. [latex]\displaystyle\int {\sin}^{456}x\cos{x}dx[/latex] or [latex]\displaystyle\int {\sin}^{2}x{\cos}^{2}xdx[/latex]

57. [latex]\displaystyle\int {\tan}^{350}x{\sec}^{2}xdx[/latex] or [latex]\displaystyle\int {\tan}^{350}x\sec{x}dx[/latex]