Problem Set: Trigonometric Substitution

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

4 - 4 \sin^{2} θ

$4 - 4{\sin}^{2}\theta$

2. $9{\sec}^{2}\theta -9$

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9 \tan^{2} θ

$9{\tan}^{2}\theta$

a^{2} + a^{2} \tan^{2} θ

${a}^{2}+{a}^{2}{\tan}^{2}\theta$

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

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a^{2} {cosh}^{2} θ

${a}^{2}{\text{cosh}}^{2}\theta$

16 {cosh}^{2} θ - 16

$16{\text{cosh}}^{2}\theta -16$

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

6. $4{x}^{2}-4x+1$

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4 {(x - \frac{1}{2})}^{2}

$4{\left(x-\frac{1}{2}\right)}^{2}$

2 x^{2} - 8 x + 3

$2{x}^{2}-8x+3$

8. $\text{-}{x}^{2}-2x+4$

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- {(x + 1)}^{2} + 5

$\text{-}{\left(x+1\right)}^{2}+5$

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

\int \frac{d x}{\sqrt{4 - x^{2}}}

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

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

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ln | x + \sqrt{- a^{2} + x^{2}} | + C

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

11.

\int \sqrt{4 - x^{2}} d x

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

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

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\frac{1}{3} ln | \sqrt{9 x^{2} + 1} + 3 x | + C

$\frac{1}{3}\text{ln}|\sqrt{9{x}^{2}+1}+3x|+C$

13.

\int \frac{x^{2} d x}{\sqrt{1 - x^{2}}}

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

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

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

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

15.

\int \frac{d x}{{(1 + x^{2})}^{2}}

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

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

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9 [\frac{x \sqrt{x^{2} + 9}}{18} + \frac{1}{2} l n | \frac{\sqrt{x^{2} + 9}}{3} + \frac{x}{3} |] + C

$9\left[\frac{x\sqrt{{x}^{2}+9}}{18}+\frac{1}{2}ln|\frac{\sqrt{{x}^{2}+9}}{3}+\frac{x}{3}|\right]+C$

17.

\int \frac{\sqrt{x^{2} - 25}}{x} d x

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

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

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- \frac{1}{3} \sqrt{9 - θ^{2}} (18 + θ^{2}) + C

$-\frac{1}{3}\sqrt{9-{\theta }^{2}}\left(18+{\theta }^{2}\right)+C$

19.

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

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

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

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\frac{(- 1 + x^{2}) (2 + 3 x^{2}) \sqrt{x^{6} - x^{8}}}{15 x^{3}} + C

$\frac{\left(-1+{x}^{2}\right)\left(2+3{x}^{2}\right)\sqrt{{x}^{6}-{x}^{8}}}{15{x}^{3}}+C$

21.

\int \frac{d x}{{(1 + x^{2})}^{\frac{3}{2}}}

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

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

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- \frac{x}{9 \sqrt{- 9 + x^{2}}} + C

$-\frac{x}{9\sqrt{-9+{x}^{2}}}+C$

23.

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

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

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

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

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

25.

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

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

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

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

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

27.

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

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

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

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\frac{1}{8} (x (5 - 2 x^{2}) \sqrt{1 - x^{2}} + 3 arcsin x) + C

$\frac{1}{8}\left(x\left(5 - 2{x}^{2}\right)\sqrt{1-{x}^{2}}+3\text{arcsin}x\right)+C$

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.

\int \frac{d x}{\sqrt{x^{2} - 1}}

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

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

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ln x - ln | 1 + \sqrt{1 - x^{2}} | + C

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

31.

\int \sqrt{x^{2} - 1} d x

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

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

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

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

33.

\int \frac{d x}{1 - x^{2}}

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

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

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- \frac{\sqrt{1 + x^{2}}}{x} + arcsinh x + C

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

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

35.

\int \frac{1}{x^{2} - 6 x} d x

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

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

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- \frac{1}{1 + x} + C

$-\frac{1}{1+x}+C$

37.

\int \frac{1}{\sqrt{- x^{2} + 2 x + 8}} d x

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

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

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\frac{2 \sqrt{- 10 + x} \sqrt{x} ln | \sqrt{- 10 + x} + \sqrt{x} |}{\sqrt{(10 - x) x}} + C

$\frac{2\sqrt{-10+x}\sqrt{x}\text{ln}|\sqrt{-10+x}+\sqrt{x}|}{\sqrt{\left(10-x\right)x}}+C$

39.

\int \frac{1}{\sqrt{x^{2} + 4 x - 12}} d x

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

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

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\frac{9 π}{2}

$\frac{9\pi }{2}$ ; area of a semicircle with radius 3

41. Find the area enclosed by the ellipse

\frac{x^{2}}{4} + \frac{y^{2}}{9} = 1

$\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$ .

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?

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arcsin (x) + C

$\text{arcsin}\left(x\right)+C$ is the common answer.

43. Evaluate the integral

\int \frac{d x}{x \sqrt{x^{2} - 1}}

$\displaystyle\int \frac{dx}{x\sqrt{{x}^{2}-1}}$ using the substitution

x = \sec θ

$x=\sec\theta$ . Next, evaluate the same integral using the substitution

x = \csc θ

$x=\csc\theta$ . 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?

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

$\frac{1}{2}\text{ln}\left(1+{x}^{2}\right)+C$ is the result using either method.

45. State the method of integration you would use to evaluate the integral

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

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

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Use trigonometric substitution. Let

x = \sec (θ)

$x=\sec\left(\theta \right)$ .

47. Evaluate

\int_{- 1}^{1} \frac{x d x}{x^{2} + 1}

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

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.

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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, and x = \sqrt{2}

$y={x}^{2},y=0,x=0,\text{and }x=\sqrt{2}$ 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.

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\frac{π^{2}}{8} + \frac{π}{4}

$\frac{{\pi }^{2}}{8}+\frac{\pi }{4}$

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

51.

(x^{2} + 36) \frac{d y}{d x} = 1, y (6) = 0

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

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

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y = \frac{1}{16} ln | \frac{x + 8}{x - 8} | + 3

$y=\frac{1}{16}\text{ln}|\frac{x+8}{x - 8}|+3$

53. Find the area bounded by

y = \frac{2}{\sqrt{64 - 4 x^{2}}}, x = 0, y = 0, and x = 2

$y=\frac{2}{\sqrt{64 - 4{x}^{2}}},x=0,y=0,\text{and }x=2$ .

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).

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55. During each cycle, the velocity v (in feet per second) of a robotic welding device is given by

v = 2 t - \frac{14}{4 + t^{2}}

$v=2t-\frac{14}{4+{t}^{2}}$ , where t is time in seconds. Find the expression for the displacement s (in feet) as a function of t if

s = 0

$s=0$ when

t = 0

$t=0$ .

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

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\frac{2 π}{3}

$\frac{2\pi }{3}$

Techniques of Integration: Supplemental Content

Problem Set: Trigonometric Substitution

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