Problem Set: Improper Integrals

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

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

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\int_{0}^{\infty} \frac{1}{4 + x^{2}} d x

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

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

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\frac{π}{2}

$\frac{\pi }{2}$

\int_{1}^{\infty} \frac{1}{x ln x} d x

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

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

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\frac{2}{e}

$\frac{2}{e}$

\int_{- \infty}^{\infty} \frac{x}{x^{2} + 1} d x

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

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 $g\left(x\right)=\frac{1}{\sqrt{{x}^{3}}}$ .

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8. Without integrating, determine whether the integral

\int_{1}^{\infty} \frac{1}{\sqrt{x + 1}} d x

${\displaystyle\int }_{1}^{\infty }\frac{1}{\sqrt{x+1}}dx$ converges or diverges.

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

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

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Converges to

\frac{1}{2}

$\frac{1}{2}$ .

10.

\int_{1}^{\infty} \frac{ln x}{x} d x

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

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

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12.

\int_{0}^{1} ln x d x

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

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

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π

$\pi$

14.

\int_{1}^{5} \frac{d x}{\sqrt{x - 1}}

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

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

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16.

\int_{0}^{\infty} e^{- x} d x

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

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

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18.

\int_{- \infty}^{\infty} \frac{e^{x}}{1 + e^{2 x}} d x

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

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

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20.

\int_{0}^{2} \frac{d x}{x^{3}}

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

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

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22.

\int_{0}^{1} \frac{d x}{\sqrt{1 - x^{2}}}

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

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

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24.

\int_{1}^{\infty} \frac{5}{x^{3}} d x

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

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

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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.

\int_{1}^{\infty} \frac{d x}{x^{2} + 4 x}

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

\int_{1}^{\infty} \frac{d x}{x^{2}}

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

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

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

28.

\int_{1}^{\infty} \frac{d x}{x^{e}}

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

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

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30.

\int_{0}^{1} \frac{d x}{\sqrt{1 - x}}

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

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

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32.

\int_{- \infty}^{0} \frac{d x}{x^{2} + 1}

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

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

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π

$\pi$

34.

\int_{0}^{1} \frac{ln x}{x} d x

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

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

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36.

\int_{0}^{\infty} x e^{- x} d x

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

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

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38.

\int_{0}^{\infty} e^{- x} d x

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

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

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40.

\int_{- 27}^{1} \frac{d x}{x^{\frac{2}{3}}}

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

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

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\frac{π}{2}

$\frac{\pi }{2}$

42.

\int_{6}^{24} \frac{d t}{t \sqrt{t^{2} - 36}}

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

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

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8 ln (16) - 4

$8\text{ln}\left(16\right)-4$

44.

\int_{0}^{3} \frac{x}{\sqrt{9 - x^{2}}} d x

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

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

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1.047

$1.047$

46. Evaluate

\int_{1}^{4} \frac{d x}{\sqrt{x^{2} - 1}}

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

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

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- 1 + \frac{2}{\sqrt{3}}

$-1+\frac{2}{\sqrt{3}}$

48. Find the area of the region in the first quadrant between the curve

y = e^{- 6 x}

$y={e}^{-6x}$ 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 $x=1$ .

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50. Find the area under the curve

y = \frac{1}{{(x + 1)}^{\frac{3}{2}}}

$y=\frac{1}{{\left(x+1\right)}^{\frac{3}{2}}}$ , bounded on the left by

x = 3

$x=3$ .

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

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\frac{5 π}{2}

$\frac{5\pi }{2}$

52. Find the volume of the solid generated by revolving about the x-axis the region under the curve

y = \frac{3}{x}

$y=\frac{3}{x}$ from

x = 1

$x=1$ to

x = \infty

$x=\infty$ .

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.

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3 π

$3\pi$

54. Find the volume of the solid generated by revolving about the x-axis the area under the curve

y = 3 e^{- x}

$y=3{e}^{\text{-}x}$ in the first quadrant.

The 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. $f\left(x\right)=1$

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\frac{1}{s}, s > 0

$\frac{1}{s},s>0$

56.

f (x) = x

$f\left(x\right)=x$

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

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\frac{s}{s^{2} + 4}, s > 0

$\frac{s}{{s}^{2}+4},s>0$

58.

f (x) = e^{a x}

$f\left(x\right)={e}^{ax}$

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

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A non-negative function is a 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 (x) = {\begin{matrix} 0 if x < 0 7 e^{- 7 x} if x \geq 0 \end{matrix}

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

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Techniques of Integration: Supplemental Content

Problem Set: Improper Integrals

Candela Citations