8.1.1: Exercises

In exercises 1 - 8, evaluate the following integrals. If the integral is not convergent, answer “It diverges.”

1) \(\displaystyle \int ^4_2\frac{dx}{(x−3)^2}\)

Answer

It diverges.

2) \(\displaystyle \int ^ \infty _0\frac{1}{4+x^2}\,dx\)

3) \(\displaystyle \int ^2_0\frac{1}{\sqrt{4−x^2}}\,dx\)

Answer

Converges to \(\frac{ \pi }{2}\)

4) \(\displaystyle \int ^ \infty _1\frac{1}{x\ln x}\,dx\)

5) \(\displaystyle \int ^ \infty _1xe^{−x}\,dx\)

Answer

Converges to \(\frac{2}{e}\)

6) \(\displaystyle \int ^ \infty _{− \infty }\frac{x}{x^2+1}\,dx\)

7) Without integrating, determine whether the integral \(\displaystyle \int ^ \infty _1\frac{1}{\sqrt{x^3+1}}\,dx\) converges or diverges by comparing the function \(f(x)=\dfrac{1}{\sqrt{x^3+1}}\) with \(g(x)=\dfrac{1}{\sqrt{x^3}}\).

Answer

It converges.

8) Without integrating, determine whether the integral \(\displaystyle \int ^ \infty _1\frac{1}{\sqrt{x+1}}\,dx\) converges or diverges.

In exercises 9 - 25, determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.

9) \(\displaystyle \int ^ \infty _0e^{−x}\cos x\,dx\)

Answer

Converges to \(\frac{1}{2}\).

10) \(\displaystyle \int ^ \infty _1\frac{\ln x}{x}\,dx\)

11) \(\displaystyle \int ^1_0\frac{\ln x}{\sqrt{x}}\,dx\)

Answer

Converges to \(-4\).

12) \(\displaystyle \int ^1_0\ln x\,dx\)

13) \(\displaystyle \int ^ \infty _{− \infty }\frac{1}{x^2+1}\,dx\)

Answer

Converges to \( \pi \).

14) \(\displaystyle \int ^5_1\frac{dx}{\sqrt{x−1}}\)

15) \(\displaystyle \int ^2_{−2}\frac{dx}{(1+x)^2}\)

Answer

It diverges.

16) \(\displaystyle \int ^ \infty _0e^{−x}\,dx\)

17) \(\displaystyle \int ^ \infty _0\sin x\,dx\)

Answer

It diverges.

18) \(\displaystyle \int ^ \infty _{− \infty }\frac{e^x}{1+e^{2x}}\,dx\)

19) \(\displaystyle \int ^1_0\frac{dx}{\sqrt[3]{x}}\)

Answer

Converges to \(1.5\).

20) \(\displaystyle \int ^2_0\frac{dx}{x^3}\)

21) \(\displaystyle \int ^2_{−1}\frac{dx}{x^3}\)

Answer

It diverges.

22) \(\displaystyle \int ^1_0\frac{dx}{\sqrt{1−x^2}}\)

23) \(\displaystyle \int ^3_0\frac{1}{x−1}\,dx\)

Answer

It diverges.

24) \(\displaystyle \int ^ \infty _1\frac{5}{x^3}\,dx\)

25) \(\displaystyle \int ^5_3\frac{5}{(x−4)^2}\,dx\)

Answer

It diverges.

In exercises 26 and 27, 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) \(\displaystyle \int ^ \infty _1\frac{dx}{x^2+4x};\) compare with \(\displaystyle \int ^ \infty _1\frac{dx}{x^2}\).

27) \(\displaystyle \int ^ \infty _1\frac{dx}{\sqrt{x}+1};\) compare with \(\displaystyle \int ^ \infty _1\frac{dx}{2\sqrt{x}}\).

Answer

Both integrals diverge.

In exercises 28 - 38, evaluate the integrals. If the integral diverges, answer “It diverges.”

28) \(\displaystyle \int ^ \infty _1\frac{dx}{x^e}\)

29) \(\displaystyle \int ^1_0\frac{dx}{x^ \pi }\)

Answer

It diverges.

30) \(\displaystyle \int ^1_0\frac{dx}{\sqrt{1−x}}\)

31) \(\displaystyle \int ^1_0\frac{dx}{1−x}\)

Answer

It diverges.

32) \(\displaystyle \int ^0_{− \infty }\frac{dx}{x^2+1}\)

33) \(\displaystyle \int ^1_{−1}\frac{dx}{\sqrt{1−x^2}}\)

Answer

Converges to \( \pi \).

34) \(\displaystyle \int ^1_0\frac{\ln x}{x}\,dx\)

35) \(\displaystyle \int ^e_0\ln(x)\,dx\)

Answer

Converges to \(0\).

36) \(\displaystyle \int ^ \infty _0xe^{−x}\,dx\)

37) \(\displaystyle \int ^ \infty _{− \infty }\frac{x}{(x^2+1)^2}\,dx\)

Answer

Converges to \(0\).

38) \(\displaystyle \int ^ \infty _0e^{−x}\,dx\)

In exercises 39 - 44, 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) \(\displaystyle \int ^9_0\frac{dx}{\sqrt{9−x}}\)

Answer

Converges to \(6\).

40) \(\displaystyle \int ^1_{−27}\frac{dx}{x^{2/3}}\)

41) \(\displaystyle \int ^3_0\frac{dx}{\sqrt{9−x^2}}\)

Answer

Converges to \(\frac{ \pi }{2}\).

42) \(\displaystyle \int ^{24}_6\frac{dt}{t\sqrt{t^2−36}}\)

43) \(\displaystyle \int ^4_0x\ln(4x)\,dx\)

Answer

Converges to \(8\ln(16)−4\).

44) \(\displaystyle \int ^3_0\frac{x}{\sqrt{9−x^2}}\,dx\)

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

Answer

Converges to about \(1.047\).

46) Evaluate \(\displaystyle \int ^4_1\frac{dx}{\sqrt{x^2−1}}.\) (Express the answer in exact form.)

47) Evaluate \(\displaystyle \int ^ \infty _2\frac{dx}{(x^2−1)^{3/2}}.\)

Answer

Converges to \(−1+\frac{2}{\sqrt{3}}\).

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 \(y=\dfrac{7}{x^2},\) the \(x\)-axis, and on the left by \(x=1.\)

Answer

\(A = 7.0\) units.²

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

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

Answer

\(A = \dfrac{5 \pi }{2}\) units.²

52) Find the volume of the solid generated by revolving about the \(x\)-axis the region under the curve \(y=\dfrac{3}{x}\) 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 \(y=6e^{−2x}\) in the first quadrant.

Answer

\(V = 3 \pi \,\text{units}^3\)

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 \([0, \infty )\) is defined by \(\displaystyle F(s)= \int ^ \infty _0e^{−sx}f(x)\,dx\) (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) \(f(x)=1\)

Answer

\(\dfrac{1}{s},\quad s>0\)

56) \(f(x)=x\)

57) \(f(x)=\cos(2x)\)

Answer

\(\dfrac{s}{s^2+4},\quad s>0\)

58) \(f(x)=e^{ax}\)

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

Answer

Answers will vary.

A function is a probability density function if it satisfies the following definition: \(\displaystyle \int ^ \infty _{− \infty }f(t)\,dt=1\). The probability that a random variable \(x\) lies between a and b is given by \(\displaystyle P(a \leq x \leq b)= \int ^b_af(t)\,dt.\)

60) Show that \(\displaystyle f(x)=\begin{cases}0,&\text{if}\,x<0\\7e^{−7x},&\text{if}\,x \geq 0\end{cases}\) 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.

Answer

0.8775