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7.4E: Exercises for Integration by Partial Fractions

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Use partial fraction decomposition (or a simpler technique) to express the rational function as a sum or difference of two or more simpler rational expressions.

1) 1(x3)(x2)

2) x2+1x(x+1)(x+2)

Answer
x2+1x(x+1)(x+2)=2x+1+52(x+2)+12x

3) 1x3x

4) 3x+1x2

Answer
3x+1x2=1x2+3x

5) 3x2x2+1 (Hint: Use long division first.)

6) 2x4x22x

Answer
2x4x22x=2x2+4x+8+16x2

7) 1(x1)(x2+1)

8) 1x2(x1)

Answer
1x2(x1)=1x21x+1x1

9) xx24

10) 1x(x1)(x2)(x3)

Answer
1x(x1)(x2)(x3)=12(x2)+12(x1)16x+16(x3)

11) 1x41=1(x+1)(x1)(x2+1)

12) 3x2x31=3x2(x1)(x2+x+1)

Answer
3x2x31=1x1+2x+1x2+x+1

13) 2x(x+2)2

14) 3x4+x3+20x2+3x+31(x+1)(x2+4)2

Answer
3x4+x3+20x2+3x+31(x+1)(x2+4)2=2x+1+xx2+41(x2+4)2

In exercises 15 - 25, use the method of partial fractions to evaluate each of the following integrals.

15) dx(x3)(x2)

16) 3xx2+2x8dx

Answer
3xx2+2x8dx=2ln|x+4|+ln|x2|+C=ln|(x+4)2(x2)|+C

17) dxx3x

18) xx24dx

Answer
Note that you don't need Partial Fractions here. We use a simple u-substitution.
xx24dx=12ln|4x2|+C

19) dxx(x1)(x2)(x3)

20) 2x2+4x+22x2+2x+10dx

Answer
Note that since the degree of the numerator is equal to the degree of the denominator, we need to start with long division.
Then note that we will need to use completing the square to continue since we cannot factor the trinomial in the denominator.
2x2+4x+22x2+2x+10dx=2(x+13arctan(1+x3))+C

21) dxx25x+6

22) 2xx2+xdx

Answer
2xx2+xdx=2ln|x|3ln|1+x|+C=ln|x2(1+x)3|+C

23) 2x2x6dx

24) dxx32x24x+8

Answer
dxx32x24x+8=116(42+xln|2+x|+ln|2+x|)+C=116(42+x+ln|x+2x2|)+C

25) dxx410x2+9

In exercises 26 - 29, evaluate the integrals with irreducible quadratic factors in the denominators.

26) 2(x4)(x2+2x+6)dx

Answer
2(x4)(x2+2x+6)dx=130(25arctan[1+x5]+2ln|4+x|ln|6+2x+x2|)+C

27) x2x3x2+4x4dx

28) x3+6x2+3x+6x3+2x2dx

Answer
Note that we need to use long division first, since the degree of the numerator is greater than the degree of the denominator.
x3+6x2+3x+6x3+2x2dx=3x+4ln|x+2|+x+C

29) x(x1)(x2+2x+2)2dx

In exercises 30 - 32, use the method of partial fractions to evaluate the integrals.

30) 3x+4(x2+4)(3x)dx

Answer
3x+4(x2+4)(3x)dx=ln|3x|+12ln|x2+4|+C

31) 2(x+2)2(2x)dx

32) 3x+4x32x4dx (Hint: Use the rational root theorem.)

Answer
3x+4x32x4dx=ln|x2|12ln|x2+2x+2|+C

In exercises 33 - 46, use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.

33) 10ex36e2xdx (Give the exact answer and the decimal equivalent. Round to five decimal places.)

34) exe2xexdx

Answer
exe2xexdx=x+ln|1ex|+C

35) sinxdx1cos2x

36) sinxcos2x+cosx6dx

Answer
sinxcos2x+cosx6dx=15ln|cosx+3cosx2|+C

37) 1x1+xdx

38) dt(etet)2

Answer
dt(etet)2=122e2t+C

39) 1+ex1exdx

40) dx1+x+1

Answer
dx1+x+1=21+x2ln|1+1+x|+C

41) dxx+4x

42) cosxsinx(1sinx)dx

Answer
cosxsinx(1sinx)dx=ln|sinx1sinx|+C

43) ex(e2x4)2dx

44) 211x24x2dx

Answer
211x24x2dx=34

45) 12+exdx

46) 11+exdx

Answer
11+exdx=xln(1+ex)+C

In exercises 47 - 48, use the given substitution to convert the integral to an integral of a rational function, then evaluate.

47) 1t3tdt;t=x3

48) 1x+3xdx;x=u6

Answer
1x+3xdx=6x1/63x1/3+2x6ln(1+x1/6)+C

49) Graph the curve y=x1+x over the interval [0,5]. Then, find the area of the region bounded by the curve, the x-axis, and the line x=4.

CNX_Calc_Figure_07_04_201.jpeg

50) Find the volume of the solid generated when the region bounded by y=1x(3x),y=0,x=1, and x=2 is revolved about the x-axis.

Answer
V=43πarctanh[13]=13πln4units3

51) The velocity of a particle moving along a line is a function of time given by v(t)=88t2t2+1. Find the distance that the particle has traveled after t=5 sec.

In exercises 52 - 54, solve the initial-value problem for x as a function of t.

52) (t27t+12)dxdt=1,t>4,x(5)=0

Answer
x=ln|t3|+ln|t4|+ln2=ln|2(t4)t3|

53) (t+5)dxdt=x2+1,t>5,x(1)=tan1

54) (2t32t2+t1)dxdt=3,x(2)=0

Answer
x=ln|t1|2arctan(2t)12ln(t2+12)+2arctan(22)+12ln4.5

55) Find the x-coordinate of the centroid of the area bounded by y(x29)=1,y=0,x=4, and x=5. (Round the answer to two decimal places.)

56) Find the volume generated by revolving the area bounded by y=1x3+7x2+6x,x=1,x=7, and y=0 about the y-axis.

Answer
V=25πln2813units3

57) Find the area bounded by y=x12x28x20,y=0,x=2, and x=4. (Round the answer to the nearest hundredth.)

58) Evaluate the integral dxx3+1.

Answer
dxx3+1=arctan[1+2x3]3+13ln|1+x|16ln1x+x2+C

For problems 59 - 62, use the substitutions tan(x2)=t, dx=21+t2dtsinx=2t1+t2, and cosx=1t21+t2.

59) dx35sinx

60) Find the area under the curve y=11+sinx between x=0 and x=π. (Assume the dimensions are in inches.)

Answer
2.0 in.2

61) Given tan(x2)=t, derive the formulas dx=21+t2dt,sinx=2t1+t2, and cosx=1t21+t2.

62) Evaluate 3x8xdx.

Answer
3x8xdx=3(8+x)1/323arctan[1+(8+x)1/33]2ln[2+(8+x)1/3]+ln[42(8+x)1/3+(8+x)2/3]+C

 


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