## Problem Set: Improper Integrals

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

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

2. ${\displaystyle\int }_{0}^{\infty }\frac{1}{4+{x}^{2}}dx$

3. ${\displaystyle\int }_{0}^{2}\frac{1}{\sqrt{4-{x}^{2}}}dx$

4. ${\displaystyle\int }_{1}^{\infty }\frac{1}{x\text{ln}x}dx$

5. ${\displaystyle\int }_{1}^{\infty }x{e}^{\text{-}x}dx$

6. ${\displaystyle\int }_{\text{-}\infty }^{\infty }\frac{x}{{x}^{2}+1}dx$

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

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

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

9. ${\displaystyle\int }_{0}^{\infty }{e}^{\text{-}x}\cos{x}dx$

10. ${\displaystyle\int }_{1}^{\infty }\frac{\text{ln}x}{x}dx$

11. ${\displaystyle\int }_{0}^{1}\frac{\text{ln}x}{\sqrt{x}}dx$

12. ${\displaystyle\int }_{0}^{1}\text{ln}xdx$

13. ${\displaystyle\int }_{\text{-}\infty }^{\infty }\frac{1}{{x}^{2}+1}dx$

14. ${\displaystyle\int }_{1}^{5}\frac{dx}{\sqrt{x - 1}}$

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

16. ${\displaystyle\int }_{0}^{\infty }{e}^{\text{-}x}dx$

17. ${\displaystyle\int }_{0}^{\infty }\sin{x}dx$

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

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

20. ${\displaystyle\int }_{0}^{2}\frac{dx}{{x}^{3}}$

21. ${\displaystyle\int }_{-1}^{2}\frac{dx}{{x}^{3}}$

22. ${\displaystyle\int }_{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}$

23. ${\displaystyle\int }_{0}^{3}\frac{1}{x - 1}dx$

24. ${\displaystyle\int }_{1}^{\infty }\frac{5}{{x}^{3}}dx$

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

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. ${\displaystyle\int }_{1}^{\infty }\frac{dx}{{x}^{2}+4x}$; compare with ${\displaystyle\int }_{1}^{\infty }\frac{dx}{{x}^{2}}$.

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

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

28. ${\displaystyle\int }_{1}^{\infty }\frac{dx}{{x}^{e}}$

29. ${\displaystyle\int }_{0}^{1}\frac{dx}{{x}^{\pi }}$

30. ${\displaystyle\int }_{0}^{1}\frac{dx}{\sqrt{1-x}}$

31. ${\displaystyle\int }_{0}^{1}\frac{dx}{1-x}$

32. ${\displaystyle\int }_{\text{-}\infty }^{0}\frac{dx}{{x}^{2}+1}$

33. ${\displaystyle\int }_{-1}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}$

34. ${\displaystyle\int }_{0}^{1}\frac{\text{ln}x}{x}dx$

35. ${\displaystyle\int }_{0}^{e}\text{ln}\left(x\right)dx$

36. ${\displaystyle\int }_{0}^{\infty }x{e}^{\text{-}x}dx$

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

38. ${\displaystyle\int }_{0}^{\infty }{e}^{\text{-}x}dx$

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. ${\displaystyle\int }_{0}^{9}\frac{dx}{\sqrt{9-x}}$

40. ${\displaystyle\int }_{-27}^{1}\frac{dx}{{x}^{\frac{2}{3}}}$

41. ${\displaystyle\int }_{0}^{3}\frac{dx}{\sqrt{9-{x}^{2}}}$

42. ${\displaystyle\int }_{6}^{24}\frac{dt}{t\sqrt{{t}^{2}-36}}$

43. ${\displaystyle\int }_{0}^{4}x\text{ln}\left(4x\right)dx$

44. ${\displaystyle\int }_{0}^{3}\frac{x}{\sqrt{9-{x}^{2}}}dx$

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

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

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

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

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

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

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

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

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

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

The Laplace transform of a continuous function over the interval $\left[0,\infty \right)$ is defined by $F\left(s\right)={\displaystyle\int }_{0}^{\infty }{e}^{\text{-}sx}f\left(x\right)dx$ (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. $f\left(x\right)=1$

56. $f\left(x\right)=x$

57. $f\left(x\right)=\cos\left(2x\right)$

58. $f\left(x\right)={e}^{ax}$

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

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

60. Show that $f\left(x\right)=\{\begin{array}{c}0\text{ if }x<0\\ 7{e}^{-7x}\text{ if }x\ge 0\end{array}$ 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.