Simplify the following expressions by writing each one using a single trigonometric function.
1. [latex]4 - 4{\sin}^{2}\theta[/latex]
2. [latex]9{\sec}^{2}\theta -9[/latex]
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[latex]9{\tan}^{2}\theta[/latex]
3. [latex]{a}^{2}+{a}^{2}{\tan}^{2}\theta[/latex]
4. [latex]{a}^{2}+{a}^{2}{\text{sinh}}^{2}\theta[/latex]
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[latex]{a}^{2}{\text{cosh}}^{2}\theta[/latex]
5. [latex]16{\text{cosh}}^{2}\theta -16[/latex]
Use the technique of completing the square to express each trinomial as the square of a binomial.
6. [latex]4{x}^{2}-4x+1[/latex]
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[latex]4{\left(x-\frac{1}{2}\right)}^{2}[/latex]
7. [latex]2{x}^{2}-8x+3[/latex]
8. [latex]\text{-}{x}^{2}-2x+4[/latex]
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[latex]\text{-}{\left(x+1\right)}^{2}+5[/latex]
Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.
9. [latex]\displaystyle\int \frac{dx}{\sqrt{4-{x}^{2}}}[/latex]
10. [latex]\displaystyle\int \frac{dx}{\sqrt{{x}^{2}-{a}^{2}}}[/latex]
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[latex]\text{ln}|x+\sqrt{\text{-}{a}^{2}+{x}^{2}}|+C[/latex]
11. [latex]\displaystyle\int \sqrt{4-{x}^{2}}dx[/latex]
12. [latex]\displaystyle\int \frac{dx}{\sqrt{1+9{x}^{2}}}[/latex]
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[latex]\frac{1}{3}\text{ln}|\sqrt{9{x}^{2}+1}+3x|+C[/latex]
13. [latex]\displaystyle\int \frac{{x}^{2}dx}{\sqrt{1-{x}^{2}}}[/latex]
14. [latex]\displaystyle\int \frac{dx}{{x}^{2}\sqrt{1-{x}^{2}}}[/latex]
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[latex]-\frac{\sqrt{1-{x}^{2}}}{x}+C[/latex]
15. [latex]\displaystyle\int \frac{dx}{{\left(1+{x}^{2}\right)}^{2}}[/latex]
16. [latex]\displaystyle\int \sqrt{{x}^{2}+9}dx[/latex]
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[latex]9\left[\frac{x\sqrt{{x}^{2}+9}}{18}+\frac{1}{2}ln|\frac{\sqrt{{x}^{2}+9}}{3}+\frac{x}{3}|\right]+C[/latex]
17. [latex]\displaystyle\int \frac{\sqrt{{x}^{2}-25}}{x}dx[/latex]
18. [latex]\displaystyle\int \frac{{\theta }^{3}d\theta }{\sqrt{9-{\theta }^{2}}}d\theta[/latex]
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[latex]-\frac{1}{3}\sqrt{9-{\theta }^{2}}\left(18+{\theta }^{2}\right)+C[/latex]
19. [latex]\displaystyle\int \frac{dx}{\sqrt{{x}^{6}-{x}^{2}}}[/latex]
20. [latex]\displaystyle\int \sqrt{{x}^{6}-{x}^{8}}dx[/latex]
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[latex]\frac{\left(-1+{x}^{2}\right)\left(2+3{x}^{2}\right)\sqrt{{x}^{6}-{x}^{8}}}{15{x}^{3}}+C[/latex]
21. [latex]\displaystyle\int \frac{dx}{{\left(1+{x}^{2}\right)}^{\frac{3}{2}}}[/latex]
22. [latex]\displaystyle\int \frac{dx}{{\left({x}^{2}-9\right)}^{\frac{3}{2}}}[/latex]
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[latex]-\frac{x}{9\sqrt{-9+{x}^{2}}}+C[/latex]
23. [latex]\displaystyle\int \frac{\sqrt{1+{x}^{2}}dx}{x}[/latex]
24. [latex]\displaystyle\int \frac{{x}^{2}dx}{\sqrt{{x}^{2}-1}}[/latex]
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[latex]\frac{1}{2}\left(\text{ln}|x+\sqrt{{x}^{2}-1}|+x\sqrt{{x}^{2}-1}\right)+C[/latex]
25. [latex]\displaystyle\int \frac{{x}^{2}dx}{{x}^{2}+4}[/latex]
26. [latex]\displaystyle\int \frac{dx}{{x}^{2}\sqrt{{x}^{2}+1}}[/latex]
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[latex]-\frac{\sqrt{1+{x}^{2}}}{x}+C[/latex]
27. [latex]\displaystyle\int \frac{{x}^{2}dx}{\sqrt{1+{x}^{2}}}[/latex]
28. [latex]{\displaystyle\int }_{-1}^{1}{\left(1-{x}^{2}\right)}^{\frac{3}{2}}dx[/latex]
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[latex]\frac{1}{8}\left(x\left(5 - 2{x}^{2}\right)\sqrt{1-{x}^{2}}+3\text{arcsin}x\right)+C[/latex]
In the following exercises, use the substitutions [latex]x=\text{sinh}\theta ,\text{cosh}\theta[/latex], or [latex]\text{tanh}\theta[/latex]. Express the final answers in terms of the variable x.
29. [latex]\displaystyle\int \frac{dx}{\sqrt{{x}^{2}-1}}[/latex]
30. [latex]\displaystyle\int \frac{dx}{x\sqrt{1-{x}^{2}}}[/latex]
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[latex]\text{ln}x-\text{ln}|1+\sqrt{1-{x}^{2}}|+C[/latex]
31. [latex]\displaystyle\int \sqrt{{x}^{2}-1}dx[/latex]
32. [latex]\displaystyle\int \frac{\sqrt{{x}^{2}-1}}{{x}^{2}}dx[/latex]
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[latex]-\frac{\sqrt{-1+{x}^{2}}}{x}+\text{ln}|x+\sqrt{-1+{x}^{2}}|+C[/latex]
33. [latex]\displaystyle\int \frac{dx}{1-{x}^{2}}[/latex]
34. [latex]\displaystyle\int \frac{\sqrt{1+{x}^{2}}}{{x}^{2}}dx[/latex]
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[latex]-\frac{\sqrt{1+{x}^{2}}}{x}+\text{arcsinh}x+C[/latex]
Use the technique of completing the square to evaluate the following integrals.
35. [latex]\displaystyle\int \frac{1}{{x}^{2}-6x}dx[/latex]
36. [latex]\displaystyle\int \frac{1}{{x}^{2}+2x+1}dx[/latex]
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[latex]-\frac{1}{1+x}+C[/latex]
37. [latex]\displaystyle\int \frac{1}{\sqrt{\text{-}{x}^{2}+2x+8}}dx[/latex]
38. [latex]\displaystyle\int \frac{1}{\sqrt{\text{-}{x}^{2}+10x}}dx[/latex]
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[latex]\frac{2\sqrt{-10+x}\sqrt{x}\text{ln}|\sqrt{-10+x}+\sqrt{x}|}{\sqrt{\left(10-x\right)x}}+C[/latex]
39. [latex]\displaystyle\int \frac{1}{\sqrt{{x}^{2}+4x - 12}}dx[/latex]
40. Evaluate the integral without using calculus: [latex]{\displaystyle\int }_{-3}^{3}\sqrt{9-{x}^{2}}dx[/latex].
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[latex]\frac{9\pi }{2}[/latex]; area of a semicircle with radius 3
41. Find the area enclosed by the ellipse [latex]\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1[/latex].
42. Evaluate the integral [latex]\displaystyle\int \frac{dx}{\sqrt{1-{x}^{2}}}[/latex] using two different substitutions. First, let [latex]x=\cos\theta[/latex] and evaluate using trigonometric substitution. Second, let [latex]x=\sin\theta[/latex] and use trigonometric substitution. Are the answers the same?
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[latex]\text{arcsin}\left(x\right)+C[/latex] is the common answer.
43. Evaluate the integral [latex]\displaystyle\int \frac{dx}{x\sqrt{{x}^{2}-1}}[/latex] using the substitution [latex]x=\sec\theta[/latex]. Next, evaluate the same integral using the substitution [latex]x=\csc\theta[/latex]. Show that the results are equivalent.
44. Evaluate the integral [latex]\displaystyle\int \frac{x}{{x}^{2}+1}dx[/latex] using the form [latex]\displaystyle\int \frac{1}{u}du[/latex]. Next, evaluate the same integral using [latex]x=\tan\theta[/latex]. Are the results the same?
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[latex]\frac{1}{2}\text{ln}\left(1+{x}^{2}\right)+C[/latex] is the result using either method.
45. State the method of integration you would use to evaluate the integral [latex]\displaystyle\int x\sqrt{{x}^{2}+1}dx[/latex]. Why did you choose this method?
46. State the method of integration you would use to evaluate the integral [latex]\displaystyle\int {x}^{2}\sqrt{{x}^{2}-1}dx[/latex]. Why did you choose this method?
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Use trigonometric substitution. Let [latex]x=\sec\left(\theta \right)[/latex].
47. Evaluate [latex]{\displaystyle\int }_{-1}^{1}\frac{xdx}{{x}^{2}+1}[/latex]
48. Find the length of the arc of the curve over the specified interval: [latex]y=\text{ln}x,\left[1,5\right][/latex]. 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 [latex]y={x}^{2},y=0,x=0,\text{and }x=\sqrt{2}[/latex] about the x-axis. (Round the answer to three decimal places).
50. The region bounded by the graph of [latex]f\left(x\right)=\frac{1}{1+{x}^{2}}[/latex] and the x-axis between [latex]x=0[/latex] and [latex]x=1[/latex] is revolved about the x-axis. Find the volume of the solid that is generated.
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[latex]\frac{{\pi }^{2}}{8}+\frac{\pi }{4}[/latex]
Solve the initial-value problem for y as a function of x.
51. [latex]\left({x}^{2}+36\right)\frac{dy}{dx}=1,y\left(6\right)=0[/latex]
52. [latex]\left(64-{x}^{2}\right)\frac{dy}{dx}=1,y\left(0\right)=3[/latex]
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[latex]y=\frac{1}{16}\text{ln}|\frac{x+8}{x - 8}|+3[/latex]
53. Find the area bounded by [latex]y=\frac{2}{\sqrt{64 - 4{x}^{2}}},x=0,y=0,\text{and }x=2[/latex].
54. An oil storage tank can be described as the volume generated by revolving the area bounded by [latex]y=\frac{16}{\sqrt{64+{x}^{2}}},x=0,y=0,x=2[/latex] 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 [latex]v=2t-\frac{14}{4+{t}^{2}}[/latex], where t is time in seconds. Find the expression for the displacement s (in feet) as a function of t if [latex]s=0[/latex] when [latex]t=0[/latex].
56. Find the length of the curve [latex]y=\sqrt{16-{x}^{2}}[/latex] between [latex]x=0[/latex] and [latex]x=2[/latex].
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[latex]\frac{2\pi }{3}[/latex]
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