## Problem Set: Trigonometric Substitution

Simplify the following expressions by writing each one using a single trigonometric function.

1. $4 - 4{\sin}^{2}\theta$

2. $9{\sec}^{2}\theta -9$

3. ${a}^{2}+{a}^{2}{\tan}^{2}\theta$

4. ${a}^{2}+{a}^{2}{\text{sinh}}^{2}\theta$

5. $16{\text{cosh}}^{2}\theta -16$

Use the technique of completing the square to express each trinomial as the square of a binomial.

6. $4{x}^{2}-4x+1$

7. $2{x}^{2}-8x+3$

8. $\text{-}{x}^{2}-2x+4$

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.

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

10. $\displaystyle\int \frac{dx}{\sqrt{{x}^{2}-{a}^{2}}}$

11. $\displaystyle\int \sqrt{4-{x}^{2}}dx$

12. $\displaystyle\int \frac{dx}{\sqrt{1+9{x}^{2}}}$

13. $\displaystyle\int \frac{{x}^{2}dx}{\sqrt{1-{x}^{2}}}$

14. $\displaystyle\int \frac{dx}{{x}^{2}\sqrt{1-{x}^{2}}}$

15. $\displaystyle\int \frac{dx}{{\left(1+{x}^{2}\right)}^{2}}$

16. $\displaystyle\int \sqrt{{x}^{2}+9}dx$

17. $\displaystyle\int \frac{\sqrt{{x}^{2}-25}}{x}dx$

18. $\displaystyle\int \frac{{\theta }^{3}d\theta }{\sqrt{9-{\theta }^{2}}}d\theta$

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

20. $\displaystyle\int \sqrt{{x}^{6}-{x}^{8}}dx$

21. $\displaystyle\int \frac{dx}{{\left(1+{x}^{2}\right)}^{\frac{3}{2}}}$

22. $\displaystyle\int \frac{dx}{{\left({x}^{2}-9\right)}^{\frac{3}{2}}}$

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

24. $\displaystyle\int \frac{{x}^{2}dx}{\sqrt{{x}^{2}-1}}$

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

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

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

28. ${\displaystyle\int }_{-1}^{1}{\left(1-{x}^{2}\right)}^{\frac{3}{2}}dx$

In the following exercises, use the substitutions $x=\text{sinh}\theta ,\text{cosh}\theta$, or $\text{tanh}\theta$. Express the final answers in terms of the variable x.

29. $\displaystyle\int \frac{dx}{\sqrt{{x}^{2}-1}}$

30. $\displaystyle\int \frac{dx}{x\sqrt{1-{x}^{2}}}$

31. $\displaystyle\int \sqrt{{x}^{2}-1}dx$

32. $\displaystyle\int \frac{\sqrt{{x}^{2}-1}}{{x}^{2}}dx$

33. $\displaystyle\int \frac{dx}{1-{x}^{2}}$

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

Use the technique of completing the square to evaluate the following integrals.

35. $\displaystyle\int \frac{1}{{x}^{2}-6x}dx$

36. $\displaystyle\int \frac{1}{{x}^{2}+2x+1}dx$

37. $\displaystyle\int \frac{1}{\sqrt{\text{-}{x}^{2}+2x+8}}dx$

38. $\displaystyle\int \frac{1}{\sqrt{\text{-}{x}^{2}+10x}}dx$

39. $\displaystyle\int \frac{1}{\sqrt{{x}^{2}+4x - 12}}dx$

40. Evaluate the integral without using calculus: ${\displaystyle\int }_{-3}^{3}\sqrt{9-{x}^{2}}dx$.

41. Find the area enclosed by the ellipse $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$.

42. Evaluate the integral $\displaystyle\int \frac{dx}{\sqrt{1-{x}^{2}}}$ using two different substitutions. First, let $x=\cos\theta$ and evaluate using trigonometric substitution. Second, let $x=\sin\theta$ and use trigonometric substitution. Are the answers the same?

43. Evaluate the integral $\displaystyle\int \frac{dx}{x\sqrt{{x}^{2}-1}}$ using the substitution $x=\sec\theta$. Next, evaluate the same integral using the substitution $x=\csc\theta$. Show that the results are equivalent.

44. Evaluate the integral $\displaystyle\int \frac{x}{{x}^{2}+1}dx$ using the form $\displaystyle\int \frac{1}{u}du$. Next, evaluate the same integral using $x=\tan\theta$. Are the results the same?

45. State the method of integration you would use to evaluate the integral $\displaystyle\int x\sqrt{{x}^{2}+1}dx$. Why did you choose this method?

46. State the method of integration you would use to evaluate the integral $\displaystyle\int {x}^{2}\sqrt{{x}^{2}-1}dx$. Why did you choose this method?

47. Evaluate ${\displaystyle\int }_{-1}^{1}\frac{xdx}{{x}^{2}+1}$

48. Find the length of the arc of the curve over the specified interval: $y=\text{ln}x,\left[1,5\right]$. Round the answer to three decimal places.

49. Find the surface area of the solid generated by revolving the region bounded by the graphs of $y={x}^{2},y=0,x=0,\text{and }x=\sqrt{2}$ about the x-axis. (Round the answer to three decimal places).

50. The region bounded by the graph of $f\left(x\right)=\frac{1}{1+{x}^{2}}$ and the x-axis between $x=0$ and $x=1$ is revolved about the x-axis. Find the volume of the solid that is generated.

Solve the initial-value problem for y as a function of x.

51. $\left({x}^{2}+36\right)\frac{dy}{dx}=1,y\left(6\right)=0$

52. $\left(64-{x}^{2}\right)\frac{dy}{dx}=1,y\left(0\right)=3$

53. Find the area bounded by $y=\frac{2}{\sqrt{64 - 4{x}^{2}}},x=0,y=0,\text{and }x=2$.

54. An oil storage tank can be described as the volume generated by revolving the area bounded by $y=\frac{16}{\sqrt{64+{x}^{2}}},x=0,y=0,x=2$ about the x-axis. Find the volume of the tank (in cubic meters).

55. During each cycle, the velocity v (in feet per second) of a robotic welding device is given by $v=2t-\frac{14}{4+{t}^{2}}$, where t is time in seconds. Find the expression for the displacement s (in feet) as a function of t if $s=0$ when $t=0$.

56. Find the length of the curve $y=\sqrt{16-{x}^{2}}$ between $x=0$ and $x=2$.