Problem Set: Other Strategies for Integration

Use a table of integrals to evaluate the following integrals.

4 \int 0 \frac{x}{\sqrt{1 + 2 x}} d x

$\underset{0}{\overset{4}{\displaystyle\int }}\frac{x}{\sqrt{1+2x}}dx$

2. $\displaystyle\int \frac{x+3}{{x}^{2}+2x+2}dx$

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\frac{1}{2} ln | x^{2} + 2 x + 2 | + 2 arctan (x + 1) + C

$\frac{1}{2}\text{ln}|{x}^{2}+2x+2|+2\text{arctan}\left(x+1\right)+C$

\int x^{3} \sqrt{1 + 2 x^{2}} d x

$\displaystyle\int {x}^{3}\sqrt{1+2{x}^{2}}dx$

4. $\displaystyle\int \frac{1}{\sqrt{{x}^{2}+6x}}dx$

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{cosh}^{- 1} (\frac{x + 3}{3}) + C

${\text{cosh}}^{-1}\left(\frac{x+3}{3}\right)+C$

\int \frac{x}{x + 1} d x

$\displaystyle\int \frac{x}{x+1}dx$

6. $\displaystyle\int x\cdot {2}^{{x}^{2}}dx$

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\frac{2^{x^{2} - 1}}{ln 2} + C

$\frac{{2}^{{x}^{2}-1}}{\text{ln}2}+C$

\int \frac{1}{4 x^{2} + 25} d x

$\displaystyle\int \frac{1}{4{x}^{2}+25}dx$

8. $\displaystyle\int \frac{dy}{\sqrt{4-{y}^{2}}}$

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arcsin (\frac{y}{2}) + C

$\text{arcsin}\left(\frac{y}{2}\right)+C$

\int {sin}^{3} (2 x) cos (2 x) d x

$\displaystyle\int {\sin}^{3}\left(2x\right)\cos\left(2x\right)dx$

10. $\displaystyle\int \csc\left(2w\right)\cot\left(2w\right)dw$

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- \frac{1}{2} csc (2 w) + C

$-\frac{1}{2}\csc\left(2w\right)+C$

11.

\int 2^{y} d y

$\displaystyle\int {2}^{y}dy$

12. ${\displaystyle\int }_{0}^{1}\frac{3xdx}{\sqrt{{x}^{2}+8}}$

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9 - 6 \sqrt{2}

$9 - 6\sqrt{2}$

13.

\int_{\frac{- 1}{4}}^{\frac{1}{4}} {sec}^{2} (π x) tan (π x) d x

${\displaystyle\int }_{\frac{-1}{4}}^{\frac{1}{4}}{\sec}^{2}\left(\pi x\right)\tan\left(\pi x\right)dx$

14. ${\displaystyle\int }_{0}^{\frac{\pi}{2}}{\tan}^{2}\left(\frac{x}{2}\right)dx$

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2 - \frac{π}{2}

$2-\frac{\pi }{2}$

15.

\int {cos}^{3} x d x

$\displaystyle\int {\cos}^{3}xdx$

16. $\displaystyle\int {\tan}^{5}\left(3x\right)dx$

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\frac{1}{12} {tan}^{4} (3 x) - \frac{1}{6} {tan}^{2} (3 x) + \frac{1}{3} ln | sec (3 x) | + C

$\frac{1}{12}{\tan}^{4}\left(3x\right)-\frac{1}{6}{\tan}^{2}\left(3x\right)+\frac{1}{3}\text{ln}|\sec\left(3x\right)|+C$

17.

\int {sin}^{2} y {cos}^{3} y d y

$\displaystyle\int {\sin}^{2}y{\cos}^{3}ydy$

Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.

18. [T] $\displaystyle\int \frac{dw}{1+\sec\left(\frac{w}{2}\right)}$

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2 cot (\frac{w}{2}) - 2 csc (\frac{w}{2}) + w + C

$2\cot\left(\frac{w}{2}\right)-2\csc\left(\frac{w}{2}\right)+w+C$

19. [T]

\int \frac{d w}{1 - cos (7 w)}

$\displaystyle\int \frac{dw}{1-\cos\left(7w\right)}$

20. [T] ${\displaystyle\int }_{0}^{t}\frac{dt}{4\cos{t}+3\sin{t}}$

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\frac{1}{5} ln | \frac{2 (5 + 4 sin t - 3 cos t)}{4 cos t + 3 sin t} |

$\frac{1}{5}\text{ln}|\frac{2\left(5+4\sin{t} - 3\cos{t}\right)}{4\cos{t}+3\sin{t}}|$

21. [T]

\int \frac{\sqrt{x^{2} - 9}}{3 x} d x

$\displaystyle\int \frac{\sqrt{{x}^{2}-9}}{3x}dx$

22. [T] $\displaystyle\int \frac{dx}{{x}^{\frac{1}{2}}+{x}^{\frac{1}{3}}}$

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6 x^{\frac{1}{6}} - 3 x^{\frac{1}{3}} + 2 \sqrt{x} - 6 ln [1 + x^{\frac{1}{6}}] + C

$6{x}^{\frac{1}{6}}-3{x}^{\frac{1}{3}}+2\sqrt{x}-6\text{ln}\left[1+{x}^{\frac{1}{6}}\right]+C$

23. [T]

\int \frac{d x}{x \sqrt{x - 1}}

$\displaystyle\int \frac{dx}{x\sqrt{x - 1}}$

24. [T] $\displaystyle\int {x}^{3}\sin{x}dx$

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- x^{3} cos x + 3 x^{2} sin x + 6 x cos x - 6 sin x + C

$\text{-}{x}^{3}\cos{x}+3{x}^{2}\sin{x}+6x\cos{x} - 6\sin{x}+C$

25. [T]

\int x \sqrt{x^{4} - 9} d x

$\displaystyle\int x\sqrt{{x}^{4}-9}dx$

26. [T] $\displaystyle\int \frac{x}{1+{e}^{\text{-}{x}^{2}}}dx$

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\frac{1}{2} (x^{2} + ln | 1 + e^{- x^{2}} |) + C

$\frac{1}{2}\left({x}^{2}+\text{ln}|1+{e}^{\text{-}{x}^{2}}|\right)+C$

27. [T]

\int \frac{\sqrt{3 - 5 x}}{2 x} d x

$\displaystyle\int \frac{\sqrt{3 - 5x}}{2x}dx$

28. [T] $\displaystyle\int \frac{dx}{x\sqrt{x - 1}}$

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2 arctan (\sqrt{x - 1}) + C

$2\text{arctan}\left(\sqrt{x - 1}\right)+C$

29. [T]

\int e^{x} {cos}^{- 1} (e^{x}) d x

$\displaystyle\int {e}^{x}{\cos}^{-1}\left({e}^{x}\right)dx$

Use a calculator or CAS to evaluate the following integrals.

30. [T] ${\displaystyle\int }_{0}^{\frac{z\pi}{4}}\cos\left(2x\right)dx$

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0.5 = \frac{1}{2}

$0.5=\frac{1}{2}$

31. [T] ${\displaystyle\int }_{0}^{1}x\cdot {e}^{\text{-}{x}^{2}}dx$

32. [T] ${\displaystyle\int }_{0}^{8}\frac{2x}{\sqrt{{x}^{2}+36}}dx$

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33. [T]

\int_{0}^{\frac{2}{\sqrt{3}}} \frac{1}{4 + 9 x^{2}} d x

${\displaystyle\int }_{0}^{\frac{2}{\sqrt{3}}}\frac{1}{4+9{x}^{2}}dx$

34. [T] $\displaystyle\int \frac{dx}{{x}^{2}+4x+13}$

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\frac{1}{3} arctan (\frac{1}{3} (x + 2)) + C

$\frac{1}{3}\text{arctan}\left(\frac{1}{3}\left(x+2\right)\right)+C$

35. [T]

\int \frac{d x}{1 + sin x}

$\displaystyle\int \frac{dx}{1+\sin{x}}$

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. $\displaystyle\int \frac{dx}{{x}^{2}+2x+10}$

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\frac{1}{3} arctan (\frac{x + 1}{3}) + C

$\frac{1}{3}\text{arctan}\left(\frac{x+1}{3}\right)+C$

37.

\int \frac{d x}{\sqrt{x^{2} - 6 x}}

$\displaystyle\int \frac{dx}{\sqrt{{x}^{2}-6x}}$

38. $\displaystyle\int \frac{{e}^{x}}{\sqrt{{e}^{2x}-4}}dx$

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$\text{ln}\left({e}^{x}+\sqrt{4+{e}^{2x}}\right)+C$

39.

$\displaystyle\int \frac{\cos{x}}{{\sin}^{2}x+2\sin{x}}dx$

40. $\displaystyle\int \frac{\text{arctan}\left({x}^{3}\right)}{{x}^{4}}dx$

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$\text{ln}x-\frac{1}{6}\text{ln}\left({x}^{6}+1\right)-\frac{\text{arctan}\left({x}^{3}\right)}{3{x}^{3}}+C$

41.

$\displaystyle\int \frac{\text{ln}|x|\text{arcsin}\left(\text{ln}|x|\right)}{x}dx$

Use tables to perform the integration.

42. $\displaystyle\int \frac{dx}{\sqrt{{x}^{2}+16}}$

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$\text{ln}|x+\sqrt{16+{x}^{2}}|+C$

43.

$\displaystyle\int \frac{3x}{2x+7}dx$

44. $\displaystyle\int \frac{dx}{1-\cos\left(4x\right)}$

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$-\frac{1}{4}\cot\left(2x\right)+C$

45.

$\displaystyle\int \frac{dx}{\sqrt{4x+1}}$

46. Find the area bounded by $y\left(4+25{x}^{2}\right)=5,x=0,y=0,\text{and }x=4$ . Use a table of integrals or a CAS.

Show Solution

$\frac{1}{2}\text{arctan}10$

47. The region bounded between the curve

$y=\frac{1}{\sqrt{1+\cos{x}}},0.3\le x\le 1.1$ , 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 $y={e}^{x},0\le x\le 3$ , about the x-axis. (Round the answer to two decimal places.)

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49. [T] Use an integral table and a calculator to find the area of the surface generated by revolving the curve

$y=\frac{{x}^{2}}{2},0\le x\le 1$ , 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 $y=\cos{x},0\le x\le \frac{\pi }{2}$ , about the x-axis. (Round the answer to two decimal places.)

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51. Find the length of the curve

$y=\frac{{x}^{2}}{4}$ over

$\left[0,8\right]$ .

52. Find the length of the curve $y={e}^{x}$ over $\left[0,\text{ln}\left(2\right)\right]$ .

Show Solution

$\sqrt{5}-\sqrt{2}+\text{ln}|\frac{2+2\sqrt{2}}{1+\sqrt{5}}|$

53. Find the area of the surface formed by revolving the graph of

$y=2\sqrt{x}$ over the interval

$\left[0,9\right]$ about the x-axis.

54. Find the average value of the function $f\left(x\right)=\frac{1}{{x}^{2}+1}$ over the interval $\left[-3,3\right]$ .

Show Solution

$\frac{1}{3}\text{arctan}\left(3\right)\approx 0.416$

55. Approximate the arc length of the curve

$y=\tan\left(\pi x\right)$ over the interval

$\left[0,\frac{1}{4}\right]$ . (Round the answer to three decimal places.)

Techniques of Integration: Supplemental Content

Problem Set: Other Strategies for Integration

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