2.6E: Exercises
- Page ID
- 18588
This page is a draft and is under active development.
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
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divergent
Exercise \(\PageIndex{2}\)
\(\displaystyle ∫^∞_0\frac{1}{4+x^2}dx\)
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Exercise \(\PageIndex{3}\)
\(\displaystyle ∫^2_0\frac{1}{\sqrt{4−x^2}}dx\)
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\(\displaystyle \frac{π}{2}\)
Exercise \(\PageIndex{4}\)
\(\displaystyle ∫^∞_1\frac{1}{xlnx}dx\)
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Exercise \(\PageIndex{5}\)
\(\displaystyle ∫^∞_1xe^{−x}dx\)
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\(\displaystyle \frac{2}{e}\)
Exercise \(\PageIndex{6}\)
\(\displaystyle ∫^∞_{−∞}\frac{x}{x^2+1}dx\)
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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
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Converges
Exercise \(\PageIndex{8}\)
Without integrating, determine whether the integral \(\displaystyle ∫^∞_1\frac{1}{\sqrt{x+1}}dx\) converges or diverges.
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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
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Converges to \(\displaystyle 1/2\).
Exercise \(\PageIndex{10}\)
\(\displaystyle ∫^∞_1\frac{lnx}{x}dx\)
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Exercise \(\PageIndex{11}\)
\(\displaystyle ∫^1_0\frac{lnx}{\sqrt{x}}dx\)
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\(\displaystyle −4\)
Exercise \(\PageIndex{12}\)
\(\displaystyle ∫^1_0lnxdx\)
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Exercise \(\PageIndex{13}\)
\(\displaystyle ∫^∞_{−∞}\frac{1}{x^2+1}dx\)
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\(\displaystyle π\)
Exercise \(\PageIndex{14}\)
\(\displaystyle ∫^5_1\frac{dx}{\sqrt{x−1}}\)
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Exercise \(\PageIndex{15}\)
\(\displaystyle ∫^2_{−2}\frac{dx}{(1+x)^2}\)
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diverges
Exercise \(\PageIndex{16}\)
\(\displaystyle ∫^∞_0e^{−x}dx\)
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Exercise \(\PageIndex{17}\)
\(\displaystyle ∫^∞_0sinxdx\)
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diverges
Exercise \(\PageIndex{18}\)
\(\displaystyle ∫^∞_{−∞}\frac{e^x}{1+e^{2x}}dx\)
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Exercise \(\PageIndex{19}\)
\(\displaystyle ∫^1_0\frac{dx}{\sqrt[3]{x}}\)
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\(\displaystyle 1.5\)
Exercise \(\PageIndex{20}\)
\(\displaystyle ∫^2_0\frac{dx}{x^3}\)
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Exercise \(\PageIndex{21}\)
\(\displaystyle ∫^2_{−1}\frac{dx}{x^3}\)
- Answer
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diverges
Exercise \(\PageIndex{22}\)
\(\displaystyle ∫^1_0\frac{dx}{\sqrt{1−x^2}}\)
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Exercise \(\PageIndex{23}\)
\(\displaystyle ∫^3_0\frac{1}{x−1}dx\)
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diverges
Exercise \(\PageIndex{24}\)
\(\displaystyle ∫^∞_1\frac{5}{x^3}dx\)
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Exercise \(\PageIndex{25}\)
\(\displaystyle ∫^5_3\frac{5}{(x−4)^2}dx\)
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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
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Both integrals diverge.
Evaluate the integrals. If the integral diverges, answer “diverges.”
Exercise \(\PageIndex{28}\)
\(\displaystyle ∫^∞_1\frac{dx}{x^e}\)
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Exercise \(\PageIndex{29}\)
\(\displaystyle ∫^1_0\frac{dx}{x^π}\)
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diverges
Exercise \(\PageIndex{30}\)
\(\displaystyle ∫^1_0\frac{dx}{\sqrt{1−x}}\)
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Exercise \(\PageIndex{31}\)
\(\displaystyle ∫^1_0\frac{dx}{1−x}\)
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diverges
Exercise \(\PageIndex{32}\)
\(\displaystyle ∫^0_{−∞}\frac{dx}{x^2+1}\)
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Exercise \(\PageIndex{33}\)
\(\displaystyle ∫^1_{−1}\frac{dx}{\sqrt{1−x^2}}\)
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\(\displaystyle π\)
Exercise \(\PageIndex{34}\)
\(\displaystyle ∫^1_0\frac{lnx}{x}dx\)
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Exercise \(\PageIndex{35}\)
\(\displaystyle ∫^e_0ln(x)dx\)
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\(\displaystyle 0.0\)
Exercise \(\PageIndex{36}\)
\(\displaystyle ∫^∞_0xe^{−x}dx\)
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Exercise \(\PageIndex{37}\)
\(\displaystyle ∫^∞_{−∞}\frac{x}{(x^2+1)^2}dx\)
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\(\displaystyle 0.0\)
Exercise \(\PageIndex{38}\)
\(\displaystyle ∫^∞_0e^{−x}dx\)
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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 \(\PageIndex{39}\)
\(\displaystyle ∫^9_0\frac{dx}{\sqrt{9−x}}\)
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\(\displaystyle 6.0\)
Exercise \(\PageIndex{40}\)
\(\displaystyle ∫^1_{−27}\frac{dx}{x^{2/3}}\)
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Exercise \(\PageIndex{41}\)
\(\displaystyle ∫^3_0\frac{dx}{\sqrt{9−x^2}}\)
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\(\displaystyle \frac{π}{2}\)
Exercise \(\PageIndex{42}\)
\(\displaystyle ∫^{24}_6\frac{dt}{t\sqrt{t^2−36}}\)
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Exercise \(\PageIndex{43}\)
\(\displaystyle ∫^4_0xln(4x)dx\)
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\(\displaystyle 8ln(16)−4\)
Exercise \(\PageIndex{44}\)
\(\displaystyle ∫^3_0\frac{x}{\sqrt{9−x^2}}dx\)
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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.)
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\(\displaystyle 1.047\)
Exercise \(\PageIndex{46}\)
Evaluate \(\displaystyle ∫^4_1\frac{dx}{\sqrt{x^2−1}}.\) (Express the answer in exact form.)
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Exercise \(\PageIndex{47}\)
Evaluate \(\displaystyle ∫^∞_2\frac{dx}{(x^2−1)^{3/2}}.\)
- Answer
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\(\displaystyle −1+\frac{2}{\sqrt{3}}\)
Exercise \(\PageIndex{48}\)
Find the area of the region in the first quadrant between the curve \(\displaystyle y=e^{−6x}\) and the x-axis.
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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
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\(\displaystyle 7.0\)
Exercise \(\PageIndex{50}\)
Find the area under the curve \(\displaystyle y=\frac{1}{(x+1)^{3/2}},\) bounded on the left by \(\displaystyle x=3.\)
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Exercise \(\PageIndex{51}\)
Find the area under \(\displaystyle y=\frac{5}{1+x^2}\) in the first quadrant.
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\(\displaystyle \frac{5π}{2}\)
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 \(\displaystyle x=∞.\)
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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.
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\(\displaystyle 3π\)
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.
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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 \(\PageIndex{55}\)
\(\displaystyle f(x)=1\)
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\(\displaystyle \frac{1}{s},s>0\)
Exercise \(\PageIndex{56}\)
\(\displaystyle f(x)=x\)
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Exercise \(\PageIndex{57}\)
\(\displaystyle f(x)=cos(2x)\)
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\(\displaystyle \frac{s}{s^2+4},s>0\)
Exercise \(\PageIndex{58}\)
\(\displaystyle f(x)=e^{ax}\)
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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.
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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.
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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.
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0.8775