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Mathematics LibreTexts

7.1E: Exercises for Integration by Parts

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In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals.

1) x3e2xdx

Answer
u=x3

2) x3ln(x)dx

3) y3cosydy

Answer
u=y3

4) x2arctanxdx

5) e3xsin(2x)dx

Answer
u=sin(2x)

In exercises 6 - 37, find the integral by using the simplest method. Not all problems require integration by parts.

6) vsinvdv

7) lnxdx (Hint: lnxdx is equivalent to 1ln(x)dx.)

Answer
lnxdx=x+xlnx+C

8) xcosxdx

9) tan1xdx

Answer
tan1xdx=xtan1x12ln(1+x2)+C

10) x2exdx

11) xsin(2x)dx

Answer
xsin(2x)dx=12xcos(2x)+14sin(2x)+C

12) xe4xdx

13) xexdx

Answer
xexdx=ex(1x)+C

14) xcos3xdx

15) x2cosxdx

Answer
x2cosxdx=2xcosx+(2+x2)sinx+C

16) xlnxdx

17) ln(2x+1)dx

Answer
ln(2x+1)dx=12(1+2x)(1+ln(1+2x))+C

18) x2e4xdx

19) exsinxdx

Answer
exsinxdx=12ex(cosx+sinx)+C

20) excosxdx

21) xex2dx

Answer
xex2dx=ex22+C

22) x2exdx

23) sin(ln(2x))dx

Answer
sin(ln(2x))dx=12xcos[ln(2x)]+12xsin[ln(2x)]+C

24) cos(lnx)dx

25) (lnx)2dx

Answer
(lnx)2dx=2x2xlnx+x(lnx)2+C

26) ln(x2)dx

27) x2lnxdx

Answer
x2lnxdx=x39+13x3lnx+C

28) sin1xdx

29) cos1(2x)dx

Answer
cos1(2x)dx=1214x2+xcos1(2x)+C

30) xarctanxdx

31) x2sinxdx

Answer
x2sinxdx=(2+x2)cosx+2xsinx+C

32) x3cosxdx

33) x3sinxdx

Answer
x3sinxdx=x(6+x2)cosx+3(2+x2)sinx+C

34) x3exdx

35) xsec1xdx

Answer
xsec1xdx=12x(11x2+xsec1x)+C

36) xsec2xdx

37) xcoshxdx

Answer
xcoshxdx=coshx+xsinhx+C

In exercises 38 - 46, compute the definite integrals. Use a graphing utility to confirm your answers.

38) 11/elnxdx

39) 10xe2xdx (Express the answer in exact form.)

Answer
10xe2xdx=1434e2

40) 10exdx(letu=x)

41) e1ln(x2)dx

Answer
e1ln(x2)dx=2

42) π0xcosxdx

43) ππxsinxdx (Express the answer in exact form.)

Answer
ππxsinxdx=2π

44) 30ln(x2+1)dx (Express the answer in exact form.)

45) π/20x2sinxdx (Express the answer in exact form.)

Answer
π/20x2sinxdx=2+π

46) 10x5xdx (Express the answer using five significant digits.)

47) Evaluate cosxln(sinx)dx

Answer
cosxln(sinx)dx=sin(x)+ln[sin(x)]sinx+C

In exercises 48 - 50, derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral.

48) xnexdx=xnexnxn1exdx

49) xncosxdx=xnsinxnxn1sinxdx

Answer
Answers vary

50) xnsinxdx=______

51) Integrate 2x2x3dx using two methods:

a. Using parts, letting dv=2x3dx

b. Substitution, letting u=2x3

Answer
a. 2x2x3dx=25(1+x)(3+2x)3/2+C
b. 2x2x3dx=25(1+x)(3+2x)3/2+C

In exercises 52 - 57, state whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem.

52) xlnxdx

53) ln2xxdx

Answer
Do not use integration by parts. Choose u to be lnx, and the integral is of the form u2du.

54) xexdx

55) xex23dx

Answer
Do not use integration by parts. Let u=x23, and the integral can be put into the form eudu.

56) x2sinxdx

57) x2sin(3x3+2)dx

Answer
Do not use integration by parts. Choose u to be u=3x3+2 and the integral can be put into the form sin(u)du.

In exercises 58-59, sketch the region bounded above by the curve, the x-axis, and x=1, and find the area of the region. Provide the exact form or round answers to the number of places indicated.

58) y=2xex (Approximate answer to four decimal places.)

59) y=exsin(πx) (Approximate answer to five decimal places.)

Answer
The area under graph is 0.39535units2.
CNX_Calc_Figure_07_01_202.jpeg

In exercises 60 - 61, find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated.

60) y=sinx,y=0,x=2π,x=3π; about the y-axis (Express the answer in exact form.)

61) y=ex,y=0,x=1,x=0; about x=1 (Express the answer in exact form.)

Answer
V=2πeunits3

62) A particle moving along a straight line has a velocity of v(t)=t2et after t sec. How far does it travel in the first 2 sec? (Assume the units are in feet and express the answer in exact form.)

63) Find the area under the graph of y=sec3x from x=0 to x=1. (Round the answer to two significant digits.)

Answer
A=2.05units2

64) Find the area between y=(x2)ex and the x-axis from x=2 to x=5. (Express the answer in exact form.)

65) Find the area of the region enclosed by the curve y=xcosx and the x-axis for 11π2x13π2. (Express the answer in exact form.)

Answer
A=12πunits2

66) Find the volume of the solid generated by revolving the region bounded by the curve y=lnx, the x-axis, and the vertical line x=e2 about the x-axis. (Express the answer in exact form.)

67) Find the volume of the solid generated by revolving the region bounded by the curve y=4cosx and the x-axis, π2x3π2, about the x-axis. (Express the answer in exact form.)

Answer
V=8π2units3

68) Find the volume of the solid generated by revolving the region in the first quadrant bounded by y=ex and the x-axis, from x=0 to x=ln(7), about the y-axis. (Express the answer in exact form.)

69) What is the volume of the Bundt cake that comes from rotating y=sinx around the y-axis from x=0 to x=π?

CNX_Calc_Figure_06_02_245.jpeg

Answer
V=2π2 units3

 


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