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Mathematics LibreTexts

8.2E: Exercises for Section 8.2

  • Gilbert Strang & Edwin “Jed” Herman
  • OpenStax

( \newcommand{\kernel}{\mathrm{null}\,}\)

Fill in the blank to make a true statement.

1) sin2x+_______=1

Answer
cos2x

2) sec2x1=_______

Answer
tan2x

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

3) sin2x=_______

Answer
1cos(2x)2

4) cos2x=_______

Answer
1+cos(2x)2

Evaluate each of the following integrals by u-substitution.

5) sin3xcosxdx

Answer
sin3xcosxdx=sin4x4+C

6) cosxsinxdx

7) tan5(x)sec2(x)dx

Answer
tan5(x)sec2(x)dx=16tan6(2x)+C

8) sin7(2x)cos(2x)dx

9) tan(x2)sec2(x2)dx

Answer
tan(x2)sec2(x2)dx=tan2(x2)+C

10) tan2xsec2xdx

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) sin3xdx

Answer
sin3xdx=3cosx4+112cos(3x)+C=cosx+cos3x3+C

12) cos3xdx

13) sinxcosxdx

Answer
sinxcosxdx=12cos2x+C

14) cos5xdx

15) sin5xcos2xdx

Answer
sin5xcos2xdx=5cosx641192cos(3x)+3320cos(5x)1448cos(7x)+C

16) sin3xcos3xdx

17) sinxcosxdx

Answer
sinxcosxdx=23(sinx)3/2+C

18) sinxcos3xdx

19) secxtanxdx

Answer
secxtanxdx=secx+C

20) tan(5x)dx

21) tan2xsecxdx

Answer
tan2xsecxdx=12secxtanx12ln(secx+tanx)+C

22) tanxsec3xdx

Answer
tanxsec3xdx=13(secx)3+C

23) sec4xdx

Answer
sec4xdx=2tanx3+13sec2xtanx=tanx+tan3x3+C

24) cotxdx

25) cscxdx

Answer
cscxdx=ln|cotx+cscx|+C

26) tan3xsecxdx

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

27) sin2axcosaxdx

Answer
sin2axcosaxdx=sin3(ax)3a+C

28) sinaxcosaxdx.

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

29) π0sin2xdx

Answer
π0sin2xdx=π2

30) π0sin4xdx

Answer
π0sin4xdx=3π8

31) cos23xdx

Answer
cos23xdx=x2+112sin(6x)+C

32) sin2xcos2xdx

33) sin2xdx+cos2xdx

Answer
sin2xdx+cos2xdx=x+C

34) sin2xcos2(2x)dx

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

35) 2π0cosxsin2xdx

Answer
2π0cosxsin2xdx=0

36) π0sin3xsin5xdx

37) π0cos(99x)sin(101x)dx

Answer
π0cos(99x)sin(101x)dx=0

38) ππcos2(3x)dx

Answer
ππcos2(3x)dx=π

39) 2π0sinxsin(2x)sin(3x)dx

Answer
2π0sinxsin(2x)sin(3x)dx=0

40) 4π0cos(x/2)sin(x/2)dx

41) π/3π/6cos3xsinxdx (Round this answer to three decimal places.)

Answer
π/3π/6cos3xsinxdx0.239

42) π/3π/3sec2x1dx

43) π/201cos(2x)dx

Answer
π/201cos(2x)dx=2

44) Find the area of the region bounded by the graphs of the equations y=sinx,y=sin3x,x=0, and x=π2.

45) Find the area of the region bounded by the graphs of the equations y=cos2x,y=sin2x,x=π4, and x=π4.

Answer
A=1unit2

46) A particle moves in a straight line with the velocity function v(t)=sin(ωt)cos2(ωt). Find its position function x=f(t) if f(0)=0.

47) Find the average value of the function f(x)=sin2xcos3x over the interval [π,π].

Answer
0

For exercises 48 - 49, solve the differential equations.

48) dydx=sin2x. The curve passes through point (0,0).

49) dydθ=sin4(πθ)

Answer
f(x)=3θ814πsin(2πθ)+132πsin(4πθ)+C

50) Find the length of the curve y=ln(cscx),forπ4xπ2.

51) Find the length of the curve y=ln(sinx),forπ3xπ2.

Answer
s=ln(3)

52) Find the volume generated by revolving the curve y=cos(3x) about the x-axis, for 0xπ36.

For exercises 53 - 54, use this information: The inner product of two functions f and g over [a,b] is defined by f(x)g(x)=f,g=bafgdx. 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
ππsin(2x)cos(3x)dx=0

54) Evaluate ππsin(mx)cos(nx)dx.

55) Integrate y=tanxsec4x.

Answer
y=tanxsec4xdx=23(tanx)3/2+27(tanx)7/2+C=221(tanx)3/2[7+3tan2x]+C

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

56) sin456xcosxdx or sin2xcos2xdx

57) tan350xsec2xdx or tan350xsecxdx

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

This page titled 8.2E: Exercises for Section 8.2 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

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