Fill in the blank to make a true statement.
1. [latex]{\sin}^{2}x+\_\_\_\_\_\_\_=1[/latex]
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[latex]{\cos}^{2}x[/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]
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[latex]\frac{1-\cos\left(2x\right)}{2}[/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]
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[latex]\frac{{\sin}^{4}x}{4}+C[/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]
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[latex]\frac{1}{12}{\tan}^{6}\left(2x\right)+C[/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]
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[latex]{\sec}^{2}\left(\frac{x}{2}\right)+C[/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]
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[latex]\text{-}\cos{x}+\frac{1}{3}\cos^{2}x+C[/latex]
12. [latex]\displaystyle\int {\cos}^{3}xdx[/latex]
13. [latex]\displaystyle\int \sin{x}\cos{x}dx[/latex]
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[latex]-\frac{1}{2}{\cos}^{2}x+C[/latex] or [latex]\frac{1}{2}{\sin}^{2}x+C[/latex]
14. [latex]\displaystyle\int {\cos}^{5}xdx[/latex]
15. [latex]\displaystyle\int {\sin}^{5}x{\cos}^{2}xdx[/latex]
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[latex]-\frac{1}{3}\cos^{3}x+\frac{2}{5}\cos^{5}x-\frac{1}{7}\cos^{7}x+C[/latex]
16. [latex]\displaystyle\int {\sin}^{3}x{\cos}^{3}xdx[/latex]
17. [latex]\displaystyle\int \sqrt{\sin{x}}\cos{x}dx[/latex]
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[latex]\frac{2}{3}{\left(\sin{x}\right)}^{\frac{3}{2}}+C[/latex]
18. [latex]\displaystyle\int \sqrt{\sin{x}}{\cos}^{3}xdx[/latex]
19. [latex]\displaystyle\int \sec{x}\tan{x}dx[/latex]
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[latex]\sec{x}+C[/latex]
20. [latex]\displaystyle\int \tan\left(5x\right)dx[/latex]
21. [latex]\displaystyle\int {\tan}^{2}x\sec{x}dx[/latex]
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[latex]\frac{1}{2}\sec{x}\tan{x}-\frac{1}{2}\text{ln}\left(\sec{x}+\tan{x}\right)+C[/latex]
22. [latex]\displaystyle\int \tan{x}{\sec}^{3}xdx[/latex]
23. [latex]\displaystyle\int {\sec}^{4}xdx[/latex]
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[latex]\frac{2\tan{x}}{3}+\frac{1}{3}\sec{\left(x\right)}^{2}\tan{x}[/latex] [latex]=\tan{x}+\frac{{\tan}^{3}x}{3}+C[/latex]
24. [latex]\displaystyle\int \cot{x}dx[/latex]
25. [latex]\displaystyle\int \csc{x}dx[/latex]
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[latex]\text{-}\text{ln}|\cot{x}+\csc{x}|+C[/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]
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[latex]\frac{{\sin}^{3}\left(ax\right)}{3a}+C[/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]
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[latex]\frac{\pi }{2}[/latex]
30. [latex]{\displaystyle\int }_{0}^{\pi }{\sin}^{4}xdx[/latex]
31. [latex]\displaystyle\int {\cos}^{2}3xdx[/latex]
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[latex]\frac{x}{2}+\frac{1}{12}\sin\left(6x\right)+C[/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]
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[latex]x+C[/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.)
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Approximately 0.239
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]
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[latex]\sqrt{2}[/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]
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[latex]\frac{3\theta }{8}-\frac{1}{4\pi }\sin\left(2\pi \theta \right)+\frac{1}{32\pi }\sin\left(4\pi \theta \right)+C=f\left(x\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].
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[latex]\text{ln}\left(\sqrt{3}\right)[/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].
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[latex]{\displaystyle\int }_{\text{-}\pi }^{\pi }\sin\left(2x\right)\cos\left(3x\right)dx=0[/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].
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[latex]\sqrt{\tan\left(x\right)}x\left(\frac{8\tan{x}}{21}+\frac{2}{7}\sec{x}^{2}\tan{x}\right)+C=f\left(x\right)[/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]
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The second integral is more difficult because the first integral is simply a u-substitution type.
