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Mathematics LibreTexts

2.6E: Exercises

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    Improper Integrals

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

    Exercise \(\PageIndex{1}\)

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

    Answer

    divergent

    Exercise \(\PageIndex{2}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{3}\)

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

    Answer

    \(\displaystyle \frac{π}{2}\)

    Exercise \(\PageIndex{4}\)

    \(\displaystyle ∫^∞_1\frac{1}{xlnx}dx\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{5}\)

    \(\displaystyle ∫^∞_1xe^{−x}dx\)

    Answer

    \(\displaystyle \frac{2}{e}\)

    Exercise \(\PageIndex{6}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{7}\)

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

    Answer

    Converges

    Exercise \(\PageIndex{8}\)

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

    Answer

    Add texts here. Do not delete this text first.

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

    Exercise \(\PageIndex{9}\)

    \(\displaystyle ∫^∞_0e^{−x}cosxdx\)

    Answer

    Converges to \(\displaystyle 1/2\).

    Exercise \(\PageIndex{10}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{11}\)

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

    Answer

    \(\displaystyle −4\)

    Exercise \(\PageIndex{12}\)

    \(\displaystyle ∫^1_0lnxdx\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{13}\)

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

    Answer

    \(\displaystyle π\)

    Exercise \(\PageIndex{14}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{15}\)

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

    Answer

    diverges

    Exercise \(\PageIndex{16}\)

    \(\displaystyle ∫^∞_0e^{−x}dx\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{17}\)

    \(\displaystyle ∫^∞_0sinxdx\)

    Answer

    diverges

    Exercise \(\PageIndex{18}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{19}\)

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

    Answer

    \(\displaystyle 1.5\)

    Exercise \(\PageIndex{20}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{21}\)

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

    Answer

    diverges

    Exercise \(\PageIndex{22}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{23}\)

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

    Answer

    diverges

    Exercise \(\PageIndex{24}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{25}\)

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

    Answer

    diverges

    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.

    Exercise \(\PageIndex{26}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{27}\)

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

    Answer

    Both integrals diverge.

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

    Exercise \(\PageIndex{28}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{29}\)

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

    Answer

    diverges

    Exercise \(\PageIndex{30}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{31}\)

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

    Answer

    diverges

    Exercise \(\PageIndex{32}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{33}\)

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

    Answer

    \(\displaystyle π\)

    Exercise \(\PageIndex{34}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{35}\)

    \(\displaystyle ∫^e_0ln(x)dx\)

    Answer

    \(\displaystyle 0.0\)

    Exercise \(\PageIndex{36}\)

    \(\displaystyle ∫^∞_0xe^{−x}dx\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{37}\)

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

    Answer

    \(\displaystyle 0.0\)

    Exercise \(\PageIndex{38}\)

    \(\displaystyle ∫^∞_0e^{−x}dx\)

    Answer

    Add texts here. Do not delete this text first.

    Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.

    Exercise \(\PageIndex{39}\)

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

    Answer

    \(\displaystyle 6.0\)

    Exercise \(\PageIndex{40}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{41}\)

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

    Answer

    \(\displaystyle \frac{π}{2}\)

    Exercise \(\PageIndex{42}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{43}\)

    \(\displaystyle ∫^4_0xln(4x)dx\)

    Answer

    \(\displaystyle 8ln(16)−4\)

    Exercise \(\PageIndex{44}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{45}\)

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

    Answer

    \(\displaystyle 1.047\)

    Exercise \(\PageIndex{46}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{47}\)

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

    Answer

    \(\displaystyle −1+\frac{2}{\sqrt{3}}\)

    Exercise \(\PageIndex{48}\)

    Find the area of the region in the first quadrant between the curve \(\displaystyle y=e^{−6x}\) and the x-axis.

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{49}\)

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

    Answer

    \(\displaystyle 7.0\)

    Exercise \(\PageIndex{50}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{51}\)

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

    Answer

    \(\displaystyle \frac{5π}{2}\)

    Exercise \(\PageIndex{52}\)

    Find the volume of the solid generated by revolving about the x-axis the region under the curve \(\displaystyle y=\frac{3}{x}\) from \(\displaystyle x=1\) to \(\displaystyle x=∞.\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{53}\)

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

    Answer

    \(\displaystyle 3π\)

    Exercise \(\PageIndex{54}\)

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

    Answer

    Add texts here. Do not delete this text first.

    The Laplace transform of a continuous function over the interval \(\displaystyle [0,∞)\) 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.

    Exercise \(\PageIndex{55}\)

    \(\displaystyle f(x)=1\)

    Answer

    \(\displaystyle \frac{1}{s},s>0\)

    Exercise \(\PageIndex{56}\)

    \(\displaystyle f(x)=x\)

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{57}\)

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

    Answer

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

    Exercise \(\PageIndex{58}\)

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

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{59}\)

    Use the formula for arc length to show that the circumference of the circle \(\displaystyle x^2+y^2=1\) 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 \(\displaystyle P(a≤x≤b)=∫^b_af(t)dt.\)

    Exercise \(\PageIndex{60}\)

    Show that \(\displaystyle f(x)=\begin{cases}0&ifx<0\\7e^{−7x}&ifx≥0\end{cases}\) is a probability density function.

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{61}\)

    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