Problem Set: Trigonometric Integrals

Fill in the blank to make a true statement.

1. ${\sin}^{2}x+\_\_\_\_\_\_\_=1$

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{cos}^{2} x

${\cos}^{2}x$

{sec}^{2} x - 1 =_______

${\sec}^{2}x - 1=\_\_\_\_\_\_\_$

Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power.

3. ${\sin}^{2}x=\_\_\_\_\_\_\_$

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\frac{1 - cos (2 x)}{2}

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

{cos}^{2} x =_______

${\cos}^{2}x=\_\_\_\_\_\_\_$

Evaluate each of the following integrals by u-substitution.

5. $\displaystyle\int {\sin}^{3}x\cos{x}dx$

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\frac{{sin}^{4} x}{4} + C

$\frac{{\sin}^{4}x}{4}+C$

\int \sqrt{cos x} sin x d x

$\displaystyle\int \sqrt{\cos{x}}\sin{x}dx$

7. $\displaystyle\int {\tan}^{5}\left(2x\right){\sec}^{2}\left(2x\right)dx$

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\frac{1}{12} {tan}^{6} (2 x) + C

$\frac{1}{12}{\tan}^{6}\left(2x\right)+C$

\int {sin}^{7} (2 x) cos (2 x) d x

$\displaystyle\int {\sin}^{7}\left(2x\right)\cos\left(2x\right)dx$

9. $\displaystyle\int \tan\left(\frac{x}{2}\right){\sec}^{2}\left(\frac{x}{2}\right)dx$

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{sec}^{2} (\frac{x}{2}) + C

${\sec}^{2}\left(\frac{x}{2}\right)+C$

10.

\int {tan}^{2} x {sec}^{2} x d x

$\displaystyle\int {\tan}^{2}x{\sec}^{2}xdx$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

11. $\displaystyle\int {\sin}^{3}xdx$

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- cos x + \frac{1}{3} {cos}^{2} x + C

$\text{-}\cos{x}+\frac{1}{3}\cos^{2}x+C$

12.

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

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

13. $\displaystyle\int \sin{x}\cos{x}dx$

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

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

\frac{1}{2} {sin}^{2} x + C

$\frac{1}{2}{\sin}^{2}x+C$

14.

\int {cos}^{5} x d x

$\displaystyle\int {\cos}^{5}xdx$

15. $\displaystyle\int {\sin}^{5}x{\cos}^{2}xdx$

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- \frac{1}{3} {cos}^{3} x + \frac{2}{5} {cos}^{5} x - \frac{1}{7} {cos}^{7} x + C

$-\frac{1}{3}\cos^{3}x+\frac{2}{5}\cos^{5}x-\frac{1}{7}\cos^{7}x+C$

16.

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

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

17. $\displaystyle\int \sqrt{\sin{x}}\cos{x}dx$

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\frac{2}{3} {(sin x)}^{\frac{3}{2}} + C

$\frac{2}{3}{\left(\sin{x}\right)}^{\frac{3}{2}}+C$

18.

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

$\displaystyle\int \sqrt{\sin{x}}{\cos}^{3}xdx$

19. $\displaystyle\int \sec{x}\tan{x}dx$

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sec x + C

$\sec{x}+C$

20.

\int tan (5 x) d x

$\displaystyle\int \tan\left(5x\right)dx$

21. $\displaystyle\int {\tan}^{2}x\sec{x}dx$

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\frac{1}{2} sec x tan x - \frac{1}{2} ln (sec x + tan x) + C

$\frac{1}{2}\sec{x}\tan{x}-\frac{1}{2}\text{ln}\left(\sec{x}+\tan{x}\right)+C$

22.

\int tan x {sec}^{3} x d x

$\displaystyle\int \tan{x}{\sec}^{3}xdx$

23. $\displaystyle\int {\sec}^{4}xdx$

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\frac{2 tan x}{3} + \frac{1}{3} sec {(x)}^{2} tan x

$\frac{2\tan{x}}{3}+\frac{1}{3}\sec{\left(x\right)}^{2}\tan{x}$

= tan x + \frac{{tan}^{3} x}{3} + C

$=\tan{x}+\frac{{\tan}^{3}x}{3}+C$

24.

\int cot x d x

$\displaystyle\int \cot{x}dx$

25. $\displaystyle\int \csc{x}dx$

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- ln | cot x + csc x | + C

$\text{-}\text{ln}|\cot{x}+\csc{x}|+C$

26.

\int \frac{{tan}^{3} x}{\sqrt{sec x}} d x

$\displaystyle\int \frac{{\tan}^{3}x}{\sqrt{\sec{x}}}dx$

For the following exercises, find a general formula for the integrals.

27. $\displaystyle\int {\sin}^{2}ax\cos{ax} dx$

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\frac{{sin}^{3} (a x)}{3 a} + C

$\frac{{\sin}^{3}\left(ax\right)}{3a}+C$

28.

\int sin a x cos a x d x

$\displaystyle\int \sin{ax} \cos{ax} dx$ .

Use the double-angle formulas to evaluate the following integrals.

29. ${\displaystyle\int }_{0}^{\pi }{\sin}^{2}xdx$

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\frac{π}{2}

$\frac{\pi }{2}$

30.

\int_{0}^{π} {sin}^{4} x d x

${\displaystyle\int }_{0}^{\pi }{\sin}^{4}xdx$

31. $\displaystyle\int {\cos}^{2}3xdx$

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\frac{x}{2} + \frac{1}{12} sin (6 x) + C

$\frac{x}{2}+\frac{1}{12}\sin\left(6x\right)+C$

32.

\int {sin}^{2} x {cos}^{2} x d x

$\displaystyle\int {\sin}^{2}x{\cos}^{2}xdx$

33. $\displaystyle\int {\sin}^{2}xdx+\displaystyle\int {\cos}^{2}xdx$

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x + C

$x+C$

34.

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

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

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.

35. ${\displaystyle\int }_{0}^{2\pi }\cos{x}\sin2xdx$

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36.

\int_{0}^{π} sin 3 x sin 5 x d x

${\displaystyle\int }_{0}^{\pi }\sin3x\sin5xdx$

37. ${\displaystyle\int }_{0}^{\pi }\cos\left(99x\right)\sin\left(101x\right)dx$

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38.

\int_{- π}^{π} {cos}^{2} (3 x) d x

${\displaystyle\int }_{\text{-}\pi }^{\pi }{\cos}^{2}\left(3x\right)dx$

39. ${\displaystyle\int }_{0}^{2\pi }\sin{x}\sin\left(2x\right)\sin\left(3x\right)dx$

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40.

\int_{0}^{4 π} cos (\frac{x}{2}) sin (\frac{x}{2}) d x

${\displaystyle\int }_{0}^{4\pi }\cos\left(\frac{x}{2}\right)\sin\left(\frac{x}{2}\right)dx$

41. ${\displaystyle\int }_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{{\cos}^{3}x}{\sqrt{\sin{x}}}dx$ (Round this answer to three decimal places.)

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42.

\int_{\frac{- π}{3}}^{\frac{π}{3}} \sqrt{{sec}^{2} x - 1} d x

${\displaystyle\int }_{\frac{\text{-}\pi}{3}}^{\frac{\pi}{3}}\sqrt{{\sec}^{2}x - 1}dx$

43. ${\displaystyle\int }_{0}^{\frac{\pi}{2}}\sqrt{1-\cos\left(2x\right)}dx$

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\sqrt{2}

$\sqrt{2}$

44. Find the area of the region bounded by the graphs of the equations

y = sin x, y = {sin}^{3} x, x = 0, and x = \frac{π}{2}

$y=\sin{x},y={\sin}^{3}x,x=0,\text{and }x=\frac{\pi }{2}$ .

45. Find the area of the region bounded by the graphs of the equations $y={\cos}^{2}x,y={\sin}^{2}x,x=-\frac{\pi }{4},\text{and }x=\frac{\pi }{4}$ .

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46. A particle moves in a straight line with the velocity function

v (t) = sin (ω t) {cos}^{2} (ω t)

$v\left(t\right)=\sin\left(\omega t\right){\cos}^{2}\left(\omega t\right)$ . Find its position function

x = f (t)

$x=f\left(t\right)$ if

f (0) = 0

$f\left(0\right)=0$ .

47. Find the average value of the function $f\left(x\right)={\sin}^{2}x{\cos}^{3}x$ over the interval $\left[\text{-}\pi ,\pi \right]$ .

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For the following exercises, solve the differential equations.

48.

\frac{d y}{d x} = {sin}^{2} x

$\frac{dy}{dx}={\sin}^{2}x$ . The curve passes through point

(0, 0)

$\left(0,0\right)$ .

49. $\frac{dy}{d\theta }={\sin}^{4}\left(\pi \theta \right)$

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\frac{3 θ}{8} - \frac{1}{4 π} sin (2 π θ) + \frac{1}{32 π} sin (4 π θ) + C = f (x)

$\frac{3\theta }{8}-\frac{1}{4\pi }\sin\left(2\pi \theta \right)+\frac{1}{32\pi }\sin\left(4\pi \theta \right)+C=f\left(x\right)$

50. Find the length of the curve

y = ln (csc x), \frac{π}{4} \leq x \leq \frac{π}{2}

$y=\text{ln}\left(\csc{x}\right),\frac{\pi }{4}\le x\le \frac{\pi }{2}$ .

51. Find the length of the curve $y=\text{ln}\left(\sin{x}\right),\frac{\pi }{3}\le x\le \frac{\pi }{2}$ .

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ln (\sqrt{3})

$\text{ln}\left(\sqrt{3}\right)$

52. Find the volume generated by revolving the curve

y = cos (3 x)

$y=\cos\left(3x\right)$ about the x-axis,

0 \leq x \leq \frac{π}{36}

$0\le x\le \frac{\pi }{36}$ .

For the following exercises, use this information: The inner product of two functions f and g over $\left[a,b\right]$ is defined by $f\left(x\right)\cdot g\left(x\right)=\langle f,g\rangle ={\displaystyle\int }_{a}^{b}f\cdot gdx$ . Two distinct functions f and g are said to be orthogonal if $\langle f,g\rangle =0$ .

53. Show that $\left\{\sin\left(2x\right),\cos\left(3x\right)\right\}$ are orthogonal over the interval $\left[\text{-}\pi ,\pi \right]$ .

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\int_{- π}^{π} sin (2 x) cos (3 x) d x = 0

${\displaystyle\int }_{\text{-}\pi }^{\pi }\sin\left(2x\right)\cos\left(3x\right)dx=0$

54. Evaluate

\int_{- π}^{π} sin (m x) cos (n x) d x

${\displaystyle\int }_{\text{-}\pi }^{\pi }\sin\left(mx\right)\cos\left(nx\right)dx$ .

55. Integrate ${y}^{\prime }=\sqrt{\tan{x}}{\sec}^{4}x$ .

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$\sqrt{\tan\left(x\right)}x\left(\frac{8\tan{x}}{21}+\frac{2}{7}\sec{x}^{2}\tan{x}\right)+C=f\left(x\right)$

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning.

56.

$\displaystyle\int {\sin}^{456}x\cos{x}dx$ or

$\displaystyle\int {\sin}^{2}x{\cos}^{2}xdx$

57. $\displaystyle\int {\tan}^{350}x{\sec}^{2}xdx$ or $\displaystyle\int {\tan}^{350}x\sec{x}dx$

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Techniques of Integration: Supplemental Content

Problem Set: Trigonometric Integrals

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