3.3E: Exercises for Section 7.2

Fill in the blank to make a true statement.

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

Answer

$\cos^2x$

2) $\sec^2x−1=$_______

Answer

$\tan^2x$

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

3) $\sin^2x=$_______

Answer

$\dfrac{1−\cos(2x)}{2}$

4) $\cos^2x=$_______

Answer

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

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

5) $\displaystyle ∫\sin^3x\cos x\,dx$

Answer

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

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

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

Answer

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

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

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

Answer

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

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) $\displaystyle ∫\sin^3x\,dx$

Answer

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

12) $\displaystyle ∫\cos^3x\,dx$

13) $\displaystyle ∫\sin x\cos x\,dx$

Answer

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

14) $\displaystyle ∫\cos^5x\,dx$

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

Answer

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

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

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

Answer

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

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

19) $\displaystyle ∫\sec x\tan x\,dx$

Answer

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

20) $\displaystyle ∫\tan(5x)\,dx$

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

Answer

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

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

23) $\displaystyle ∫\sec^4x\,dx$

Answer

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

24) $\displaystyle ∫\cot x\,dx$

25) $\displaystyle ∫\csc x\,dx$

Answer

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

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

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

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

Answer

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

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

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

29) $\displaystyle ∫^π_0\sin^2x\,dx$

Answer

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

30) $\displaystyle ∫^π_0\sin^4 x\,dx$

31) $\displaystyle ∫\cos^2 3x\,dx$

Answer

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

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

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

Answer

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

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

Answer

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

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

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

Answer

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

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

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

Answer

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

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.)

Answer

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

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

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

Answer

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

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 $x=\frac{π}{4}.$

Answer

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

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 $[−π,π].$

Answer

$0$

For exercises 48 - 49, solve the differential equations.

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

49) $\dfrac{dy}{dθ}=\sin^4(πθ)$

Answer

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

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 $y=\ln(\sin x),\, \text{for}\,\tfrac{π}{3}≤x≤\tfrac{π}{2}.$

Answer

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

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 $[−π,\, π]$.

Answer

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

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

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

Answer

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

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 $\displaystyle ∫\tan^{350}x\sec x\,dx$

Answer

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