## Problem Set: Partial Fractions

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

1. $\frac{1}{\left(x - 3\right)\left(x - 2\right)}$

2. $\frac{{x}^{2}+1}{x\left(x+1\right)\left(x+2\right)}$

3. $\frac{1}{{x}^{3}-x}$

4. $\frac{3x+1}{{x}^{2}}$

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

6. $\frac{2{x}^{4}}{{x}^{2}-2x}$

7. $\frac{1}{\left(x - 1\right)\left({x}^{2}+1\right)}$

8. $\frac{1}{{x}^{2}\left(x - 1\right)}$

9. $\frac{x}{{x}^{2}-4}$

10. $\frac{1}{x\left(x - 1\right)\left(x - 2\right)\left(x - 3\right)}$

11. $\frac{1}{{x}^{4}-1}=\frac{1}{\left(x+1\right)\left(x - 1\right)\left({x}^{2}+1\right)}$

12. $\frac{3{x}^{2}}{{x}^{3}-1}=\frac{3{x}^{2}}{\left(x - 1\right)\left({x}^{2}+x+1\right)}$

13. $\frac{2x}{{\left(x+2\right)}^{2}}$

14. $\frac{3{x}^{4}+{x}^{3}+20{x}^{2}+3x+31}{\left(x+1\right){\left({x}^{2}+4\right)}^{2}}$

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

15. $\displaystyle\int \frac{dx}{\left(x - 3\right)\left(x - 2\right)}$

16. $\displaystyle\int \frac{3x}{{x}^{2}+2x - 8}dx$

17. $\displaystyle\int \frac{dx}{{x}^{3}-x}$

18. $\displaystyle\int \frac{x}{{x}^{2}-4}dx$

19. $\displaystyle\int \frac{dx}{x\left(x - 1\right)\left(x - 2\right)\left(x - 3\right)}$

20. $\displaystyle\int \frac{2{x}^{2}+4x+22}{{x}^{2}+2x+10}dx$

21. $\displaystyle\int \frac{dx}{{x}^{2}-5x+6}$

22. $\displaystyle\int \frac{2-x}{{x}^{2}+x}dx$

23. $\displaystyle\int \frac{2}{{x}^{2}-x - 6}dx$

24. $\displaystyle\int \frac{dx}{{x}^{3}-2{x}^{2}-4x+8}$

25. $\displaystyle\int \frac{dx}{{x}^{4}-10{x}^{2}+9}$

Evaluate the following integrals, which have irreducible quadratic factors.

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

27. $\displaystyle\int \frac{{x}^{2}}{{x}^{3}-{x}^{2}+4x - 4}dx$

28. $\displaystyle\int \frac{{x}^{3}+6{x}^{2}+3x+6}{{x}^{3}+2{x}^{2}}dx$

29. $\displaystyle\int \frac{x}{\left(x - 1\right){\left({x}^{2}+2x+2\right)}^{2}}dx$

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

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

31. $\displaystyle\int \frac{2}{{\left(x+2\right)}^{2}\left(2-x\right)}dx$

32. $\displaystyle\int \frac{3x+4}{{x}^{3}-2x - 4}dx$ (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. ${\displaystyle\int }_{0}^{1}\frac{{e}^{x}}{36-{e}^{2x}}dx$ (Give the exact answer and the decimal equivalent. Round to five decimal places.)

34. $\displaystyle\int \frac{{e}^{x}dx}{{e}^{2x}-{e}^{x}}dx$

35. $\displaystyle\int \frac{\sin{x}dx}{1-{\cos}^{2}x}$

36. $\displaystyle\int \frac{\sin{x}}{{\cos}^{2}x+\cos{x} - 6}dx$

37. $\displaystyle\int \frac{1-\sqrt{x}}{1+\sqrt{x}}dx$

38. $\displaystyle\int \frac{dt}{{\left({e}^{t}-{e}^{\text{-}t}\right)}^{2}}$

39. $\displaystyle\int \frac{1+{e}^{x}}{1-{e}^{x}}dx$

40. $\displaystyle\int \frac{dx}{1+\sqrt{x+1}}$

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

42. $\displaystyle\int \frac{\cos{x}}{\sin{x}\left(1-\sin{x}\right)}dx$

43. $\displaystyle\int \frac{{e}^{x}}{{\left({e}^{2x}-4\right)}^{2}}dx$

44. $\underset{1}{\overset{2}{\displaystyle\int }}\frac{1}{{x}^{2}\sqrt{4-{x}^{2}}}dx$

45. $\displaystyle\int \frac{1}{2+{e}^{\text{-}x}}dx$

46. $\displaystyle\int \frac{1}{1+{e}^{x}}dx$

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

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

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

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

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

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

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

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

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

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

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

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

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

58. Evaluate the integral $\displaystyle\int \frac{dx}{{x}^{3}+1}$.

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

59. $\displaystyle\int \frac{dx}{3 - 5\sin{x}}$

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

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

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