Problem Set: Improper Integrals

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

1. 24dx(x3)2

2. 014+x2dx

3. 0214x2dx

4. 11xlnxdx

5. 1xe-xdx

6. -xx2+1dx

7. Without integrating, determine whether the integral 11x3+1dx converges or diverges by comparing the function f(x)=1x3+1 with g(x)=1x3.

8. Without integrating, determine whether the integral 11x+1dx converges or diverges.

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

9. 0e-xcosxdx

10. 1lnxxdx

11. 01lnxxdx

12. 01lnxdx

13. -1x2+1dx

14. 15dxx1

15. 22dx(1+x)2

16. 0e-xdx

17. 0sinxdx

18. -ex1+e2xdx

19. 01dxx3

20. 02dxx3

21. 12dxx3

22. 01dx1x2

23. 031x1dx

24. 15x3dx

25. 355(x4)2dx

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. 1dxx2+4x; compare with 1dxx2.

27. 1dxx+1; compare with 1dx2x.

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

28. 1dxxe

29. 01dxxπ

30. 01dx1x

31. 01dx1x

32. -0dxx2+1

33. 11dx1x2

34. 01lnxxdx

35. 0eln(x)dx

36. 0xe-xdx

37. -x(x2+1)2dx

38. 0e-xdx

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. 09dx9x

40. 271dxx23

41. 03dx9x2

42. 624dttt236

43. 04xln(4x)dx

44. 03x9x2dx

45. Evaluate .51dx1x2. (Be careful!) (Express your answer using three decimal places.)

46. Evaluate 14dxx21. (Express the answer in exact form.)

47. Evaluate 2dx(x21)32.

48. Find the area of the region in the first quadrant between the curve y=e6x and the x-axis.

49. Find the area of the region bounded by the curve y=7x2, the x-axis, and on the left by x=1.

50. Find the area under the curve y=1(x+1)32, bounded on the left by x=3.

51. Find the area under y=51+x2 in the first quadrant.

52. Find the volume of the solid generated by revolving about the x-axis the region under the curve y=3x from x=1 to x=.

53. Find the volume of the solid generated by revolving about the y-axis the region under the curve y=6e2x in the first quadrant.

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

The Laplace transform of a continuous function over the interval [0,) is defined by F(s)=0e-sxf(x)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(x)=1

56. f(x)=x

57. f(x)=cos(2x)

58. f(x)=eax

59. Use the formula for arc length to show that the circumference of the circle x2+y2=1 is 2π.

A non-negative function is a probability density function if it satisfies the following definition: -f(t)dt=1. The probability that a random variable x lies between a and b is given by P(axb)=abf(t)dt.

60. Show that f(x)={0 if x<07e7x if x0 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.