# 8.2E: Exercises

• Gilbert Strang & Edwin “Jed” Herman
• OpenStax

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Fill in the blank to make a true statement.

1) $$\sin^2x+$$_______$$=1$$

$$\cos^2x$$

2) $$\sec^2x−1=$$_______

$$\tan^2x$$

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

3) $$\sin^2x=$$_______

$$\dfrac{1−\cos(2x)}{2}$$

4) $$\cos^2x=$$_______

$$\dfrac{1+\cos(2x)}{2}$$

Evaluate each of the following integrals by $$u$$-substitution.

5) $$\displaystyle ∫\sin^3x\cos x\,dx$$

$$\displaystyle ∫\sin^3x\cos x\,dx \quad = \quad \frac{\sin^4x}{4}+C$$

6) $$\displaystyle ∫\sqrt{\cos x}\sin x\,dx$$

7) $$\displaystyle ∫\tan^5(2x)\sec^2(2x)\,dx$$

$$\displaystyle ∫\tan^5(2x)\sec^2(2x)\,dx \quad = \quad \tfrac{1}{12}\tan^6(2x)+C$$

8) $$\displaystyle ∫\sin^7(2x)\cos(2x)\,dx$$

9) $$\displaystyle ∫\tan(\frac{x}{2})\sec^2(\frac{x}{2})\,dx$$

$$\displaystyle ∫\tan(\frac{x}{2})\sec^2(\frac{x}{2})\,dx \quad = \quad \tan^2(\frac{x}{2})+C$$

10) $$\displaystyle ∫\tan^2x\sec^2x\,dx$$

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) $$\displaystyle ∫\sin^3x\,dx$$

$$\displaystyle ∫\sin^3x\,dx \quad = \quad −\frac{3\cos x}{4}+\tfrac{1}{12}\cos(3x)+C=−\cos x+\frac{\cos^3x}{3}+C$$

12) $$\displaystyle ∫\cos^3x\,dx$$

13) $$\displaystyle ∫\sin x\cos x\,dx$$

$$\displaystyle ∫\sin x\cos x\,dx \quad = \quad −\tfrac{1}{2}\cos^2x+C$$

14) $$\displaystyle ∫\cos^5x\,dx$$

15) $$\displaystyle ∫\sin^5x\cos^2x\,dx$$

$$\displaystyle ∫\sin^5x\cos^2x\,dx \quad = \quad −\frac{5\cos x}{64}−\tfrac{1}{192}\cos(3x)+\tfrac{3}{320}\cos(5x)−\tfrac{1}{448}\cos(7x)+C$$

16) $$\displaystyle ∫\sin^3x\cos^3x\,dx$$

17) $$\displaystyle ∫\sqrt{\sin x}\cos x\,dx$$

$$\displaystyle ∫\sqrt{\sin x}\cos x\,dx \quad = \quad \tfrac{2}{3}(\sin x)^{3/2}+C$$

18) $$\displaystyle ∫\sqrt{\sin x}\cos^3x\,dx$$

19) $$\displaystyle ∫\sec x\tan x\,dx$$

$$\displaystyle ∫\sec x\tan x\,dx \quad = \quad \sec x+C$$

20) $$\displaystyle ∫\tan(5x)\,dx$$

21) $$\displaystyle ∫\tan^2x\sec x\,dx$$

$$\displaystyle ∫\tan^2x\sec x\,dx \quad = \quad \tfrac{1}{2}\sec x\tan x−\tfrac{1}{2}\ln(\sec x+\tan x)+C$$

22) $$\displaystyle ∫\tan x\sec^3x\,dx$$

23) $$\displaystyle ∫\sec^4x\,dx$$

$$\displaystyle ∫\sec^4x\,dx \quad = \quad \frac{2\tan x}{3}+\tfrac{1}{3}\sec^2 x\tan x=\tan x+\frac{\tan^3x}{3}+C$$

24) $$\displaystyle ∫\cot x\,dx$$

25) $$\displaystyle ∫\csc x\,dx$$

$$\displaystyle ∫\csc x\,dx \quad = \quad −\ln|\cot x+\csc x|+C$$

26) $$\displaystyle ∫\frac{\tan^3x}{\sqrt{\sec x}}\,dx$$

For exercises 27 - 28, find a general formula for the integrals.

27) $$\displaystyle ∫\sin^2ax\cos ax\,dx$$

$$\displaystyle ∫\sin^2ax\cos ax\,dx \quad = \quad \frac{\sin^3(ax)}{3a}+C$$

28) $$\displaystyle ∫\sin ax\cos ax\,dx.$$

Use the double-angle formulas to evaluate the integrals in exercises 29 - 34.

29) $$\displaystyle ∫^π_0\sin^2x\,dx$$

$$\displaystyle ∫^π_0\sin^2x\,dx \quad = \quad \frac{π}{2}$$

30) $$\displaystyle ∫^π_0\sin^4 x\,dx$$

31) $$\displaystyle ∫\cos^2 3x\,dx$$

$$\displaystyle ∫\cos^2 3x\,dx \quad = \quad \frac{x}{2}+\tfrac{1}{12}\sin(6x)+C$$

32) $$\displaystyle ∫\sin^2x\cos^2x\,dx$$

33) $$\displaystyle ∫\sin^2x\,dx+∫\cos^2x\,dx$$

$$\displaystyle ∫\sin^2x\,dx+∫\cos^2x\,dx \quad = \quad x+C$$

34) $$\displaystyle ∫\sin^2 x\cos^2(2x)\,dx$$

For exercises 35 - 43, evaluate the definite integrals. Express answers in exact form whenever possible.

35) $$\displaystyle ∫^{2π}_0\cos x\sin 2x\,dx$$

$$\displaystyle ∫^{2π}_0\cos x\sin 2x\,dx \quad = \quad 0$$

36) $$\displaystyle ∫^π_0\sin 3x\sin 5x\,dx$$

37) $$\displaystyle ∫^π_0\cos(99x)\sin(101x)\,dx$$

$$\displaystyle ∫^π_0\cos(99x)\sin(101x)\,dx \quad = \quad 0$$

38) $$\displaystyle ∫^π_{−π}\cos^2(3x)\,dx$$

39) $$\displaystyle ∫^{2π}_0\sin x\sin(2x)\sin(3x)\,dx$$

$$\displaystyle ∫^{2π}_0\sin x\sin(2x)\sin(3x)\,dx \quad = \quad 0$$

40) $$\displaystyle ∫^{4π}_0\cos(x/2)\sin(x/2)\,dx$$

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

$$\displaystyle ∫^{π/3}_{π/6}\frac{\cos^3x}{\sqrt{\sin x}}\,dx \quad \approx \quad 0.239$$

42) $$\displaystyle ∫^{π/3}_{−π/3}\sqrt{\sec^2x−1}\,dx$$

43) $$\displaystyle ∫^{π/2}_0\sqrt{1−\cos(2x)}\,dx$$

$$\displaystyle ∫^{π/2}_0\sqrt{1−\cos(2x)}\,dx \quad = \quad \sqrt{2}$$

44) Find the area of the region bounded by the graphs of the equations $$y=\sin x,\, y=\sin^3x,\, x=0,$$ and $$x=\frac{π}{2}.$$

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

$$A = 1 \,\text{unit}^2$$

46) A particle moves in a straight line with the velocity function $$v(t)=\sin(ωt)\cos^2(ωt).$$ Find its position function $$x=f(t)$$ if $$f(0)=0.$$

47) Find the average value of the function $$f(x)=\sin^2x\cos^3x$$ over the interval $$[−π,π].$$

$$0$$

For exercises 48 - 49, solve the differential equations.

48) $$\dfrac{dy}{\,dx}=\sin^2x.$$ The curve passes through point $$(0,0).$$

49) $$\dfrac{dy}{dθ}=\sin^4(πθ)$$

$$f(x) = \dfrac{3θ}{8}−\tfrac{1}{4π}\sin(2πθ)+\tfrac{1}{32π}\sin(4πθ)+C$$

50) Find the length of the curve $$y=\ln(\csc x),\, \text{for}\,\tfrac{π}{4}≤x≤\tfrac{π}{2}.$$

51) Find the length of the curve $$y=\ln(\sin x),\, \text{for}\,\tfrac{π}{3}≤x≤\tfrac{π}{2}.$$

$$s = \ln(\sqrt{3})$$

52) Find the volume generated by revolving the curve $$y=\cos(3x)$$ about the $$x$$-axis, for $$0≤x≤\tfrac{π}{36}.$$

For exercises 53 - 54, use this information: The inner product of two functions $$f$$ and $$g$$ over $$[a,b]$$ is defined by $$\displaystyle f(x)⋅g(x)=⟨f,g⟩=∫^b_af⋅g\,dx$$. Two distinct functions $$f$$ and $$g$$ are said to be orthogonal if $$⟨f,g⟩=0$$.

53) Show that $${\sin(2x),\, \cos(3x)}$$ are orthogonal over the interval $$[−π,\, π]$$.

$$\displaystyle ∫^π_{−π}\sin(2x)\cos(3x)\,dx=0$$

54) Evaluate $$\displaystyle ∫^π_{−π}\sin(mx)\cos(nx)\,dx.$$

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

$$\displaystyle y = \int \sqrt{\tan x}\sec^4x \, dx \quad = \quad \tfrac{2}{3}\left(\tan x\right)^{3/2} + \tfrac{2}{7}\left(\tan x\right)^{7/2}+C= \tfrac{2}{21}\left(\tan x\right)^{3/2}\left[ 7 + 3\tan^2 x \right]+C$$

For each pair of integrals in exercises 56 - 57, determine which one is more difficult to evaluate. Explain your reasoning.

56) $$\displaystyle ∫\sin^{456}x\cos x\,dx$$ or $$\displaystyle ∫\sin^2x\cos^2x\,dx$$

57) $$\displaystyle ∫\tan^{350}x\sec^2x\,dx$$ or $$\displaystyle ∫\tan^{350}x\sec x\,dx$$

The second integral is more difficult because the first integral is simply a $$u$$-substitution type.