2.6E: Exercises

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

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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

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Exercise \(\PageIndex{5}\)

\(\displaystyle ∫^∞_1xe^{−x}dx\)

Answer

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

Exercise \(\PageIndex{6}\)

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

Answer

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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

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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

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Exercise \(\PageIndex{11}\)

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

Answer

\(\displaystyle −4\)

Exercise \(\PageIndex{12}\)

\(\displaystyle ∫^1_0lnxdx\)

Answer

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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

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Exercise \(\PageIndex{15}\)

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

Answer

diverges

Exercise \(\PageIndex{16}\)

\(\displaystyle ∫^∞_0e^{−x}dx\)

Answer

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Exercise \(\PageIndex{17}\)

\(\displaystyle ∫^∞_0sinxdx\)

Answer

diverges

Exercise \(\PageIndex{18}\)

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

Answer

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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

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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

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Exercise \(\PageIndex{23}\)

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

Answer

diverges

Exercise \(\PageIndex{24}\)

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

Answer

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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

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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

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Exercise \(\PageIndex{29}\)

\(\displaystyle ∫^1_0\frac{dx}{x^π}\)

Answer

diverges

Exercise \(\PageIndex{30}\)

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

Answer

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Exercise \(\PageIndex{31}\)

\(\displaystyle ∫^1_0\frac{dx}{1−x}\)

Answer

diverges

Exercise \(\PageIndex{32}\)

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

Answer

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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

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Exercise \(\PageIndex{35}\)

\(\displaystyle ∫^e_0ln(x)dx\)

Answer

\(\displaystyle 0.0\)

Exercise \(\PageIndex{36}\)

\(\displaystyle ∫^∞_0xe^{−x}dx\)

Answer

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Exercise \(\PageIndex{37}\)

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

Answer

\(\displaystyle 0.0\)

Exercise \(\PageIndex{38}\)

\(\displaystyle ∫^∞_0e^{−x}dx\)

Answer

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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