## Problem Set: Other Strategies for Integration

Use a table of integrals to evaluate the following integrals.

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

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

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

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

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

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

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

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

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

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

11. $\displaystyle\int {2}^{y}dy$

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

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

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

15. $\displaystyle\int {\cos}^{3}xdx$

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

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

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

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

19. [T] $\displaystyle\int \frac{dw}{1-\cos\left(7w\right)}$

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

21. [T] $\displaystyle\int \frac{\sqrt{{x}^{2}-9}}{3x}dx$

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

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

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

25. [T] $\displaystyle\int x\sqrt{{x}^{4}-9}dx$

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

27. [T] $\displaystyle\int \frac{\sqrt{3 - 5x}}{2x}dx$

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

29. [T] $\displaystyle\int {e}^{x}{\cos}^{-1}\left({e}^{x}\right)dx$

Use a calculator or CAS to evaluate the following integrals.

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

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

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

33. [T] ${\displaystyle\int }_{0}^{\frac{2}{\sqrt{3}}}\frac{1}{4+9{x}^{2}}dx$

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

35. [T] $\displaystyle\int \frac{dx}{1+\sin{x}}$

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

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

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

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

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

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

Use tables to perform the integration.

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

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

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

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.

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.)

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.)

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 $\left[0,\text{ln}\left(2\right)\right]$.

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 $\left[-3,3\right]$.

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.)