Use a table of integrals to evaluate the following integrals.
1. [latex]\underset{0}{\overset{4}{\displaystyle\int }}\frac{x}{\sqrt{1+2x}}dx[/latex]
2. [latex]\displaystyle\int \frac{x+3}{{x}^{2}+2x+2}dx[/latex]
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[latex]\frac{1}{2}\text{ln}|{x}^{2}+2x+2|+2\text{arctan}\left(x+1\right)+C[/latex]
3. [latex]\displaystyle\int {x}^{3}\sqrt{1+2{x}^{2}}dx[/latex]
4. [latex]\displaystyle\int \frac{1}{\sqrt{{x}^{2}+6x}}dx[/latex]
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[latex]{\text{cosh}}^{-1}\left(\frac{x+3}{3}\right)+C[/latex]
5. [latex]\displaystyle\int \frac{x}{x+1}dx[/latex]
6. [latex]\displaystyle\int x\cdot {2}^{{x}^{2}}dx[/latex]
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[latex]\frac{{2}^{{x}^{2}-1}}{\text{ln}2}+C[/latex]
7. [latex]\displaystyle\int \frac{1}{4{x}^{2}+25}dx[/latex]
8. [latex]\displaystyle\int \frac{dy}{\sqrt{4-{y}^{2}}}[/latex]
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[latex]\text{arcsin}\left(\frac{y}{2}\right)+C[/latex]
9. [latex]\displaystyle\int {\sin}^{3}\left(2x\right)\cos\left(2x\right)dx[/latex]
10. [latex]\displaystyle\int \csc\left(2w\right)\cot\left(2w\right)dw[/latex]
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[latex]-\frac{1}{2}\csc\left(2w\right)+C[/latex]
11. [latex]\displaystyle\int {2}^{y}dy[/latex]
12. [latex]{\displaystyle\int }_{0}^{1}\frac{3xdx}{\sqrt{{x}^{2}+8}}[/latex]
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[latex]9 - 6\sqrt{2}[/latex]
13. [latex]{\displaystyle\int }_{\frac{-1}{4}}^{\frac{1}{4}}{\sec}^{2}\left(\pi x\right)\tan\left(\pi x\right)dx[/latex]
14. [latex]{\displaystyle\int }_{0}^{\frac{\pi}{2}}{\tan}^{2}\left(\frac{x}{2}\right)dx[/latex]
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[latex]2-\frac{\pi }{2}[/latex]
15. [latex]\displaystyle\int {\cos}^{3}xdx[/latex]
16. [latex]\displaystyle\int {\tan}^{5}\left(3x\right)dx[/latex]
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[latex]\frac{1}{12}{\tan}^{4}\left(3x\right)-\frac{1}{6}{\tan}^{2}\left(3x\right)+\frac{1}{3}\text{ln}|\sec\left(3x\right)|+C[/latex]
17. [latex]\displaystyle\int {\sin}^{2}y{\cos}^{3}ydy[/latex]
Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.
18. [T] [latex]\displaystyle\int \frac{dw}{1+\sec\left(\frac{w}{2}\right)}[/latex]
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[latex]2\cot\left(\frac{w}{2}\right)-2\csc\left(\frac{w}{2}\right)+w+C[/latex]
19. [T] [latex]\displaystyle\int \frac{dw}{1-\cos\left(7w\right)}[/latex]
20. [T] [latex]{\displaystyle\int }_{0}^{t}\frac{dt}{4\cos{t}+3\sin{t}}[/latex]
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[latex]\frac{1}{5}\text{ln}|\frac{2\left(5+4\sin{t} - 3\cos{t}\right)}{4\cos{t}+3\sin{t}}|[/latex]
21. [T] [latex]\displaystyle\int \frac{\sqrt{{x}^{2}-9}}{3x}dx[/latex]
22. [T] [latex]\displaystyle\int \frac{dx}{{x}^{\frac{1}{2}}+{x}^{\frac{1}{3}}}[/latex]
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[latex]6{x}^{\frac{1}{6}}-3{x}^{\frac{1}{3}}+2\sqrt{x}-6\text{ln}\left[1+{x}^{\frac{1}{6}}\right]+C[/latex]
23. [T] [latex]\displaystyle\int \frac{dx}{x\sqrt{x - 1}}[/latex]
24. [T] [latex]\displaystyle\int {x}^{3}\sin{x}dx[/latex]
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[latex]\text{-}{x}^{3}\cos{x}+3{x}^{2}\sin{x}+6x\cos{x} - 6\sin{x}+C[/latex]
25. [T] [latex]\displaystyle\int x\sqrt{{x}^{4}-9}dx[/latex]
26. [T] [latex]\displaystyle\int \frac{x}{1+{e}^{\text{-}{x}^{2}}}dx[/latex]
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[latex]\frac{1}{2}\left({x}^{2}+\text{ln}|1+{e}^{\text{-}{x}^{2}}|\right)+C[/latex]
27. [T] [latex]\displaystyle\int \frac{\sqrt{3 - 5x}}{2x}dx[/latex]
28. [T] [latex]\displaystyle\int \frac{dx}{x\sqrt{x - 1}}[/latex]
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[latex]2\text{arctan}\left(\sqrt{x - 1}\right)+C[/latex]
29. [T] [latex]\displaystyle\int {e}^{x}{\cos}^{-1}\left({e}^{x}\right)dx[/latex]
Use a calculator or CAS to evaluate the following integrals.
30. [T] [latex]{\displaystyle\int }_{0}^{\frac{z\pi}{4}}\cos\left(2x\right)dx[/latex]
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[latex]0.5=\frac{1}{2}[/latex]
31. [T] [latex]{\displaystyle\int }_{0}^{1}x\cdot {e}^{\text{-}{x}^{2}}dx[/latex]
32. [T] [latex]{\displaystyle\int }_{0}^{8}\frac{2x}{\sqrt{{x}^{2}+36}}dx[/latex]
33. [T] [latex]{\displaystyle\int }_{0}^{\frac{2}{\sqrt{3}}}\frac{1}{4+9{x}^{2}}dx[/latex]
34. [T] [latex]\displaystyle\int \frac{dx}{{x}^{2}+4x+13}[/latex]
Show Solution
[latex]\frac{1}{3}\text{arctan}\left(\frac{1}{3}\left(x+2\right)\right)+C[/latex]
35. [T] [latex]\displaystyle\int \frac{dx}{1+\sin{x}}[/latex]
Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table.
36. [latex]\displaystyle\int \frac{dx}{{x}^{2}+2x+10}[/latex]
Show Solution
[latex]\frac{1}{3}\text{arctan}\left(\frac{x+1}{3}\right)+C[/latex]
37. [latex]\displaystyle\int \frac{dx}{\sqrt{{x}^{2}-6x}}[/latex]
38. [latex]\displaystyle\int \frac{{e}^{x}}{\sqrt{{e}^{2x}-4}}dx[/latex]
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[latex]\text{ln}\left({e}^{x}+\sqrt{4+{e}^{2x}}\right)+C[/latex]
39. [latex]\displaystyle\int \frac{\cos{x}}{{\sin}^{2}x+2\sin{x}}dx[/latex]
40. [latex]\displaystyle\int \frac{\text{arctan}\left({x}^{3}\right)}{{x}^{4}}dx[/latex]
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[latex]\text{ln}x-\frac{1}{6}\text{ln}\left({x}^{6}+1\right)-\frac{\text{arctan}\left({x}^{3}\right)}{3{x}^{3}}+C[/latex]
41. [latex]\displaystyle\int \frac{\text{ln}|x|\text{arcsin}\left(\text{ln}|x|\right)}{x}dx[/latex]
Use tables to perform the integration.
42. [latex]\displaystyle\int \frac{dx}{\sqrt{{x}^{2}+16}}[/latex]
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[latex]\text{ln}|x+\sqrt{16+{x}^{2}}|+C[/latex]
43. [latex]\displaystyle\int \frac{3x}{2x+7}dx[/latex]
44. [latex]\displaystyle\int \frac{dx}{1-\cos\left(4x\right)}[/latex]
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[latex]-\frac{1}{4}\cot\left(2x\right)+C[/latex]
45. [latex]\displaystyle\int \frac{dx}{\sqrt{4x+1}}[/latex]
46. Find the area bounded by [latex]y\left(4+25{x}^{2}\right)=5,x=0,y=0,\text{and }x=4[/latex]. Use a table of integrals or a CAS.
Show Solution
[latex]\frac{1}{2}\text{arctan}10[/latex]
47. The region bounded between the curve [latex]y=\frac{1}{\sqrt{1+\cos{x}}},0.3\le x\le 1.1[/latex], and the x-axis is revolved about the x-axis to generate a solid. Use a table of integrals to find the volume of the solid generated. (Round the answer to two decimal places.)
48. Use substitution and a table of integrals to find the area of the surface generated by revolving the curve [latex]y={e}^{x},0\le x\le 3[/latex], about the x-axis. (Round the answer to two decimal places.)
49. [T] Use an integral table and a calculator to find the area of the surface generated by revolving the curve [latex]y=\frac{{x}^{2}}{2},0\le x\le 1[/latex], about the x-axis. (Round the answer to two decimal places.)
50. [T] Use a CAS or tables to find the area of the surface generated by revolving the curve [latex]y=\cos{x},0\le x\le \frac{\pi }{2}[/latex], about the x-axis. (Round the answer to two decimal places.)
51. Find the length of the curve [latex]y=\frac{{x}^{2}}{4}[/latex] over [latex]\left[0,8\right][/latex].
52. Find the length of the curve [latex]y={e}^{x}[/latex] over [latex]\left[0,\text{ln}\left(2\right)\right][/latex].
Show Solution
[latex]\sqrt{5}-\sqrt{2}+\text{ln}|\frac{2+2\sqrt{2}}{1+\sqrt{5}}|[/latex]
53. Find the area of the surface formed by revolving the graph of [latex]y=2\sqrt{x}[/latex] over the interval [latex]\left[0,9\right][/latex] about the x-axis.
54. Find the average value of the function [latex]f\left(x\right)=\frac{1}{{x}^{2}+1}[/latex] over the interval [latex]\left[-3,3\right][/latex].
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[latex]\frac{1}{3}\text{arctan}\left(3\right)\approx 0.416[/latex]
55. Approximate the arc length of the curve [latex]y=\tan\left(\pi x\right)[/latex] over the interval [latex]\left[0,\frac{1}{4}\right][/latex]. (Round the answer to three decimal places.)
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