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8.7E: Exercises

  • Page ID
    137771
    • Gilbert Strang & Edwin “Jed” Herman
    • OpenStax

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

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

    Answer
    It diverges.

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

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

    Answer
    Converges to \(\frac{π}{2}\)

    4) \(\displaystyle ∫^∞_1\frac{1}{x\ln x}\,dx\)

    5) \(\displaystyle ∫^∞_1xe^{−x}\,dx\)

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

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

    7) Without integrating, determine whether the integral \(\displaystyle ∫^∞_1\frac{1}{\sqrt{x^3+1}}\,dx\) 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 \(\displaystyle ∫^∞_1\frac{1}{\sqrt{x+1}}\,dx\) 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) \(\displaystyle ∫^∞_0e^{−x}\cos x\,dx\)

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

    10) \(\displaystyle ∫^∞_1\frac{\ln x}{x}\,dx\)

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

    Answer
    Converges to \(-4\).

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

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

    Answer
    Converges to \(π\).

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

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

    Answer
    It diverges.

    16) \(\displaystyle ∫^∞_0e^{−x}\,dx\)

    17) \(\displaystyle ∫^∞_0\sin x\,dx\)

    Answer
    It diverges.

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

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

    Answer
    Converges to \(1.5\).

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

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

    Answer
    It diverges.

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

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

    Answer
    It diverges.

    24) \(\displaystyle ∫^∞_1\frac{5}{x^3}\,dx\)

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

    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) \(\displaystyle ∫^∞_1\frac{dx}{x^2+4x};\) compare with \(\displaystyle ∫^∞_1\frac{dx}{x^2}\).

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

    Answer
    Both integrals diverge.

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

    28) \(\displaystyle ∫^∞_1\frac{dx}{x^e}\)

    29) \(\displaystyle ∫^1_0\frac{dx}{x^π}\)

    Answer
    It diverges.

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

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

    Answer
    It diverges.

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

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

    Answer
    Converges to \(π\).

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

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

    Answer
    Converges to \(0\).

    36) \(\displaystyle ∫^∞_0xe^{−x}\,dx\)

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

    Answer
    Converges to \(0\).

    38) \(\displaystyle ∫^∞_0e^{−x}\,dx\)

    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) \(\displaystyle ∫^9_0\frac{dx}{\sqrt{9−x}}\)

    Answer
    Converges to \(6\).

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

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

    Answer
    Converges to \(\frac{π}{2}\).

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

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

    Answer
    Converges to \(8\ln(16)−4\).

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

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

    Answer
    Converges to about \(1.047\).

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

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

    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
    \(A = \dfrac{5π}{2}\) 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 \(x=∞.\)

    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
    \(V = 3π\,\text{units}^3\)

    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 \([0,∞)\) is defined by \(\displaystyle F(s)=∫^∞_0e^{−sx}f(x)\,dx\) (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 \(2π\).

    Answer
    Answers will vary.

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

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

    This page titled 8.7E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.