Problem Set: Improper Integrals

Evaluate the following integrals. If the integral is not convergent, answer “divergent.”

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

2. [latex]{\displaystyle\int }_{0}^{\infty }\frac{1}{4+{x}^{2}}dx[/latex]

3. [latex]{\displaystyle\int }_{0}^{2}\frac{1}{\sqrt{4-{x}^{2}}}dx[/latex]

4. [latex]{\displaystyle\int }_{1}^{\infty }\frac{1}{x\text{ln}x}dx[/latex]

5. [latex]{\displaystyle\int }_{1}^{\infty }x{e}^{\text{-}x}dx[/latex]

6. [latex]{\displaystyle\int }_{\text{-}\infty }^{\infty }\frac{x}{{x}^{2}+1}dx[/latex]

7. Without integrating, determine whether the integral [latex]{\displaystyle\int }_{1}^{\infty }\frac{1}{\sqrt{{x}^{3}+1}}dx[/latex] converges or diverges by comparing the function [latex]f\left(x\right)=\frac{1}{\sqrt{{x}^{3}+1}}[/latex] with [latex]g\left(x\right)=\frac{1}{\sqrt{{x}^{3}}}[/latex].

8. Without integrating, determine whether the integral [latex]{\displaystyle\int }_{1}^{\infty }\frac{1}{\sqrt{x+1}}dx[/latex] converges or diverges.

Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.

9. [latex]{\displaystyle\int }_{0}^{\infty }{e}^{\text{-}x}\cos{x}dx[/latex]

10. [latex]{\displaystyle\int }_{1}^{\infty }\frac{\text{ln}x}{x}dx[/latex]

11. [latex]{\displaystyle\int }_{0}^{1}\frac{\text{ln}x}{\sqrt{x}}dx[/latex]

12. [latex]{\displaystyle\int }_{0}^{1}\text{ln}xdx[/latex]

13. [latex]{\displaystyle\int }_{\text{-}\infty }^{\infty }\frac{1}{{x}^{2}+1}dx[/latex]

14. [latex]{\displaystyle\int }_{1}^{5}\frac{dx}{\sqrt{x - 1}}[/latex]

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

16. [latex]{\displaystyle\int }_{0}^{\infty }{e}^{\text{-}x}dx[/latex]

17. [latex]{\displaystyle\int }_{0}^{\infty }\sin{x}dx[/latex]

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

19. [latex]{\displaystyle\int }_{0}^{1}\frac{dx}{\sqrt[3]{x}}[/latex]

20. [latex]{\displaystyle\int }_{0}^{2}\frac{dx}{{x}^{3}}[/latex]

21. [latex]{\displaystyle\int }_{-1}^{2}\frac{dx}{{x}^{3}}[/latex]

22. [latex]{\displaystyle\int }_{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}[/latex]

23. [latex]{\displaystyle\int }_{0}^{3}\frac{1}{x - 1}dx[/latex]

24. [latex]{\displaystyle\int }_{1}^{\infty }\frac{5}{{x}^{3}}dx[/latex]

25. [latex]{\displaystyle\int }_{3}^{5}\frac{5}{{\left(x - 4\right)}^{2}}dx[/latex]

Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.

26. [latex]{\displaystyle\int }_{1}^{\infty }\frac{dx}{{x}^{2}+4x}[/latex]; compare with [latex]{\displaystyle\int }_{1}^{\infty }\frac{dx}{{x}^{2}}[/latex].

27. [latex]{\displaystyle\int }_{1}^{\infty }\frac{dx}{\sqrt{x}+1}[/latex]; compare with [latex]{\displaystyle\int }_{1}^{\infty }\frac{dx}{2\sqrt{x}}[/latex].

Evaluate the integrals. If the integral diverges, answer “diverges.”

28. [latex]{\displaystyle\int }_{1}^{\infty }\frac{dx}{{x}^{e}}[/latex]

29. [latex]{\displaystyle\int }_{0}^{1}\frac{dx}{{x}^{\pi }}[/latex]

30. [latex]{\displaystyle\int }_{0}^{1}\frac{dx}{\sqrt{1-x}}[/latex]

31. [latex]{\displaystyle\int }_{0}^{1}\frac{dx}{1-x}[/latex]

32. [latex]{\displaystyle\int }_{\text{-}\infty }^{0}\frac{dx}{{x}^{2}+1}[/latex]

33. [latex]{\displaystyle\int }_{-1}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}[/latex]

34. [latex]{\displaystyle\int }_{0}^{1}\frac{\text{ln}x}{x}dx[/latex]

35. [latex]{\displaystyle\int }_{0}^{e}\text{ln}\left(x\right)dx[/latex]

36. [latex]{\displaystyle\int }_{0}^{\infty }x{e}^{\text{-}x}dx[/latex]

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

38. [latex]{\displaystyle\int }_{0}^{\infty }{e}^{\text{-}x}dx[/latex]

Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.

39. [latex]{\displaystyle\int }_{0}^{9}\frac{dx}{\sqrt{9-x}}[/latex]

40. [latex]{\displaystyle\int }_{-27}^{1}\frac{dx}{{x}^{\frac{2}{3}}}[/latex]

41. [latex]{\displaystyle\int }_{0}^{3}\frac{dx}{\sqrt{9-{x}^{2}}}[/latex]

42. [latex]{\displaystyle\int }_{6}^{24}\frac{dt}{t\sqrt{{t}^{2}-36}}[/latex]

43. [latex]{\displaystyle\int }_{0}^{4}x\text{ln}\left(4x\right)dx[/latex]

44. [latex]{\displaystyle\int }_{0}^{3}\frac{x}{\sqrt{9-{x}^{2}}}dx[/latex]

45. Evaluate [latex]{\displaystyle\int }_{.5}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}[/latex]. (Be careful!) (Express your answer using three decimal places.)

46. Evaluate [latex]{\displaystyle\int }_{1}^{4}\frac{dx}{\sqrt{{x}^{2}-1}}[/latex]. (Express the answer in exact form.)

47. Evaluate [latex]{\displaystyle\int }_{2}^{\infty }\frac{dx}{{\left({x}^{2}-1\right)}^{\frac{3}{2}}}[/latex].

48. Find the area of the region in the first quadrant between the curve [latex]y={e}^{-6x}[/latex] and the x-axis.

49. Find the area of the region bounded by the curve [latex]y=\frac{7}{{x}^{2}}[/latex], the x-axis, and on the left by [latex]x=1[/latex].

50. Find the area under the curve [latex]y=\frac{1}{{\left(x+1\right)}^{\frac{3}{2}}}[/latex], bounded on the left by [latex]x=3[/latex].

51. Find the area under [latex]y=\frac{5}{1+{x}^{2}}[/latex] in the first quadrant.

52. Find the volume of the solid generated by revolving about the x-axis the region under the curve [latex]y=\frac{3}{x}[/latex] from [latex]x=1[/latex] to [latex]x=\infty[/latex].

53. Find the volume of the solid generated by revolving about the y-axis the region under the curve [latex]y=6{e}^{-2x}[/latex] in the first quadrant.

54. Find the volume of the solid generated by revolving about the x-axis the area under the curve [latex]y=3{e}^{\text{-}x}[/latex] in the first quadrant.

The Laplace transform of a continuous function over the interval [latex]\left[0,\infty \right)[/latex] is defined by [latex]F\left(s\right)={\displaystyle\int }_{0}^{\infty }{e}^{\text{-}sx}f\left(x\right)dx[/latex] (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F.

55. [latex]f\left(x\right)=1[/latex]

56. [latex]f\left(x\right)=x[/latex]

57. [latex]f\left(x\right)=\cos\left(2x\right)[/latex]

58. [latex]f\left(x\right)={e}^{ax}[/latex]

59. Use the formula for arc length to show that the circumference of the circle [latex]{x}^{2}+{y}^{2}=1[/latex] is [latex]2\pi[/latex].

A non-negative function is a probability density function if it satisfies the following definition: [latex]{\displaystyle\int }_{\text{-}\infty }^{\infty }f\left(t\right)dt=1[/latex]. The probability that a random variable x lies between a and b is given by [latex]P\left(a\le x\le b\right)={\displaystyle\int }_{a}^{b}f\left(t\right)dt[/latex].

60. Show that [latex]f\left(x\right)=\{\begin{array}{c}0\text{ if }x<0\\ 7{e}^{-7x}\text{ if }x\ge 0\end{array}[/latex] is a probability density function.

61. Find the probability that x is between 0 and 0.3. (Use the function defined in the preceding problem.) Use four-place decimal accuracy.