Problem Set: Partial Fractions

Express the rational function as a sum or difference of two simpler rational expressions.

1. [latex]\frac{1}{\left(x - 3\right)\left(x - 2\right)}[/latex]

2. [latex]\frac{{x}^{2}+1}{x\left(x+1\right)\left(x+2\right)}[/latex]

3. [latex]\frac{1}{{x}^{3}-x}[/latex]

4. [latex]\frac{3x+1}{{x}^{2}}[/latex]

5. [latex]\frac{3{x}^{2}}{{x}^{2}+1}[/latex] (Hint: Use long division first.)

6. [latex]\frac{2{x}^{4}}{{x}^{2}-2x}[/latex]

7. [latex]\frac{1}{\left(x - 1\right)\left({x}^{2}+1\right)}[/latex]

8. [latex]\frac{1}{{x}^{2}\left(x - 1\right)}[/latex]

9. [latex]\frac{x}{{x}^{2}-4}[/latex]

10. [latex]\frac{1}{x\left(x - 1\right)\left(x - 2\right)\left(x - 3\right)}[/latex]

11. [latex]\frac{1}{{x}^{4}-1}=\frac{1}{\left(x+1\right)\left(x - 1\right)\left({x}^{2}+1\right)}[/latex]

12. [latex]\frac{3{x}^{2}}{{x}^{3}-1}=\frac{3{x}^{2}}{\left(x - 1\right)\left({x}^{2}+x+1\right)}[/latex]

13. [latex]\frac{2x}{{\left(x+2\right)}^{2}}[/latex]

14. [latex]\frac{3{x}^{4}+{x}^{3}+20{x}^{2}+3x+31}{\left(x+1\right){\left({x}^{2}+4\right)}^{2}}[/latex]

Use the method of partial fractions to evaluate each of the following integrals.

15. [latex]\displaystyle\int \frac{dx}{\left(x - 3\right)\left(x - 2\right)}[/latex]

16. [latex]\displaystyle\int \frac{3x}{{x}^{2}+2x - 8}dx[/latex]

17. [latex]\displaystyle\int \frac{dx}{{x}^{3}-x}[/latex]

18. [latex]\displaystyle\int \frac{x}{{x}^{2}-4}dx[/latex]

19. [latex]\displaystyle\int \frac{dx}{x\left(x - 1\right)\left(x - 2\right)\left(x - 3\right)}[/latex]

20. [latex]\displaystyle\int \frac{2{x}^{2}+4x+22}{{x}^{2}+2x+10}dx[/latex]

21. [latex]\displaystyle\int \frac{dx}{{x}^{2}-5x+6}[/latex]

22. [latex]\displaystyle\int \frac{2-x}{{x}^{2}+x}dx[/latex]

23. [latex]\displaystyle\int \frac{2}{{x}^{2}-x - 6}dx[/latex]

24. [latex]\displaystyle\int \frac{dx}{{x}^{3}-2{x}^{2}-4x+8}[/latex]

25. [latex]\displaystyle\int \frac{dx}{{x}^{4}-10{x}^{2}+9}[/latex]

Evaluate the following integrals, which have irreducible quadratic factors.

26. [latex]\displaystyle\int \frac{2}{\left(x - 4\right)\left({x}^{2}+2x+6\right)}dx[/latex]

27. [latex]\displaystyle\int \frac{{x}^{2}}{{x}^{3}-{x}^{2}+4x - 4}dx[/latex]

28. [latex]\displaystyle\int \frac{{x}^{3}+6{x}^{2}+3x+6}{{x}^{3}+2{x}^{2}}dx[/latex]

29. [latex]\displaystyle\int \frac{x}{\left(x - 1\right){\left({x}^{2}+2x+2\right)}^{2}}dx[/latex]

Use the method of partial fractions to evaluate the following integrals.

30. [latex]\displaystyle\int \frac{3x+4}{\left({x}^{2}+4\right)\left(3-x\right)}dx[/latex]

31. [latex]\displaystyle\int \frac{2}{{\left(x+2\right)}^{2}\left(2-x\right)}dx[/latex]

32. [latex]\displaystyle\int \frac{3x+4}{{x}^{3}-2x - 4}dx[/latex] (Hint: Use the rational root theorem.)

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.

33. [latex]{\displaystyle\int }_{0}^{1}\frac{{e}^{x}}{36-{e}^{2x}}dx[/latex] (Give the exact answer and the decimal equivalent. Round to five decimal places.)

34. [latex]\displaystyle\int \frac{{e}^{x}dx}{{e}^{2x}-{e}^{x}}dx[/latex]

35. [latex]\displaystyle\int \frac{\sin{x}dx}{1-{\cos}^{2}x}[/latex]

36. [latex]\displaystyle\int \frac{\sin{x}}{{\cos}^{2}x+\cos{x} - 6}dx[/latex]

37. [latex]\displaystyle\int \frac{1-\sqrt{x}}{1+\sqrt{x}}dx[/latex]

38. [latex]\displaystyle\int \frac{dt}{{\left({e}^{t}-{e}^{\text{-}t}\right)}^{2}}[/latex]

39. [latex]\displaystyle\int \frac{1+{e}^{x}}{1-{e}^{x}}dx[/latex]

40. [latex]\displaystyle\int \frac{dx}{1+\sqrt{x+1}}[/latex]

41. [latex]\displaystyle\int \frac{dx}{\sqrt{x}+\sqrt[4]{x}}[/latex]

42. [latex]\displaystyle\int \frac{\cos{x}}{\sin{x}\left(1-\sin{x}\right)}dx[/latex]

43. [latex]\displaystyle\int \frac{{e}^{x}}{{\left({e}^{2x}-4\right)}^{2}}dx[/latex]

44. [latex]\underset{1}{\overset{2}{\displaystyle\int }}\frac{1}{{x}^{2}\sqrt{4-{x}^{2}}}dx[/latex]

45. [latex]\displaystyle\int \frac{1}{2+{e}^{\text{-}x}}dx[/latex]

46. [latex]\displaystyle\int \frac{1}{1+{e}^{x}}dx[/latex]

Use the given substitution to convert the integral to an integral of a rational function, then evaluate.

47. [latex]\displaystyle\int \frac{1}{t-\sqrt[3]{t}}dtt={x}^{3}[/latex]

48. [latex]\displaystyle\int \frac{1}{\sqrt{x}+\sqrt[3]{x}}dx;x={u}^{6}[/latex]

49. Graph the curve [latex]y=\frac{x}{1+x}[/latex] over the interval [latex]\left[0,5\right][/latex]. Then, find the area of the region bounded by the curve, the x-axis, and the line [latex]x=4[/latex].


This figure is a graph of the function y = x/(1 + x). The graph is only in the first quadrant. It begins at the origin and increases into the first quadrant. The curve stops at x = 5.

50. Find the volume of the solid generated when the region bounded by [latex]y=\frac{1}{\sqrt{x\left(3-x\right)}}[/latex], [latex]y=0[/latex], [latex]x=1[/latex], and [latex]x=2[/latex] is revolved about the x-axis.

51. The velocity of a particle moving along a line is a function of time given by [latex]v\left(t\right)=\frac{88{t}^{2}}{{t}^{2}+1}[/latex]. Find the distance that the particle has traveled after [latex]t=5[/latex] sec.

Solve the initial-value problem for x as a function of t.

52. [latex]\left({t}^{2}-7t+12\right)\frac{dx}{dt}=1,\left(t>4,x\left(5\right)=0\right)[/latex]

53. [latex]\left(t+5\right)\frac{dx}{dt}={x}^{2}+1,t>\text{-}5,x\left(1\right)=\tan1[/latex]

54. [latex]\left(2{t}^{3}-2{t}^{2}+t - 1\right)\frac{dx}{dt}=3,x\left(2\right)=0[/latex]

55. Find the x-coordinate of the centroid of the area bounded by [latex]y\left({x}^{2}-9\right)=1[/latex], [latex]y=0,x=4,\text{and }x=5[/latex]. (Round the answer to two decimal places.)

56. Find the volume generated by revolving the area bounded by [latex]y=\frac{1}{{x}^{3}+7{x}^{2}+6x}x=1,x=7,\text{and }y=0[/latex] about the y-axis.

57. Find the area bounded by [latex]y=\frac{x - 12}{{x}^{2}-8x - 20}[/latex], [latex]y=0,x=2,\text{and }x=4[/latex]. (Round the answer to the nearest hundredth.)

58. Evaluate the integral [latex]\displaystyle\int \frac{dx}{{x}^{3}+1}[/latex].

For the following problems, use the substitutions [latex]\tan\left(\frac{x}{2}\right)=t[/latex], [latex]dx=\frac{2}{1+{t}^{2}}dt[/latex], [latex]\sin{x}=\frac{2t}{1+{t}^{2}}[/latex], and [latex]\cos{x}=\frac{1-{t}^{2}}{1+{t}^{2}}[/latex].

59. [latex]\displaystyle\int \frac{dx}{3 - 5\sin{x}}[/latex]

60. Find the area under the curve [latex]y=\frac{1}{1+\sin{x}}[/latex] between [latex]x=0[/latex] and [latex]x=\pi [/latex]. (Assume the dimensions are in inches.)

61. Given [latex]\tan\left(\frac{x}{2}\right)=t[/latex], derive the formulas [latex]dx=\frac{2}{1+{t}^{2}}dt[/latex], [latex]\sin{x}=\frac{2t}{1+{t}^{2}}[/latex], and [latex]\cos{x}=\frac{1-{t}^{2}}{1+{t}^{2}}[/latex].

62. Evaluate [latex]\displaystyle\int \frac{\sqrt[3]{x - 8}}{x}dx[/latex].