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Concepts and Vocabulary
Multiple Choice If a function \(f\) is continuous on the interval \([a,\,\infty) \), then \( \int_{a}^{\infty }f(x)\,dx\) is called [(a) a definite,(b) an infinite, (c) an improper, (d) a proper ] integral.
Multiple Choice If the \(\lim\limits_{b \rightarrow \infty }\int_{a}^{b} f(x)\,dx\) does not exist, the improper integral \(\int_{a}^{\infty }f(x)\,dx\) [ (a) converges, (b) diverges, (c) equals \(ab].\)
True or False If a function \(f\) is continuous for all \(x\), then the improper integral \(\int_{-\infty }^{\,\infty }f(x)\,dx\) always converges.
True or False If a function \(f\) is continuous and nonnegative on the interval \([a,\,\infty) \), and \( \int_{a}^{\infty } f( x) \,dx\) converges, then \( \int_{a}^{\infty }f(x)\,dx\) represents the area under the graph of \(y=f(x)\) for \(x \ge a.\)
True or False If a function \(f\) is continuous for all \(x,\) then the improper integral \(\int_{-\infty }^{\infty }f(x)\,dx=\lim\limits_{a\,\rightarrow \,\infty }\int_{-a}^{a}f(x)\,dx\).
To determine whether the improper integral \( \int_{a}^{b} f( x) \,dx\) converges or diverges, where \(f\) is continuous on \([a,\,b),\) but is not defined at \(b\), requires finding what limit?
Skill Building
In Problems 7–14, determine whether each integral is improper. For those that are improper, state the reason.
\(\int_{0}^{\infty }x^{2}\,dx\)
\(\int_{0}^{5}x^{3}dx\)
\(\int_{2}^{3}\dfrac{dx}{x-1}\)
\(\int_{1}^{2}\dfrac{dx}{x-1}\)
\(\int_{0}^{1}\dfrac{1}{x}\,dx\)
\(\int_{-1}^{1}\dfrac{x\, dx}{x^{2}+1}\)
\(\int_{0}^{1}\dfrac{x}{x^{2}-1}\,dx\)
\(\int_{0}^{\infty }e^{-2x}\,dx\)
In Problems 15–24, determine whether each improper integral converges or diverges. If it converges, find its value.
\(\int_{1}^{\infty }\dfrac{dx}{x^{3}}\)
\(\int_{-\infty }^{-10}\dfrac{dx}{x^{2}}\)
\(\int_{0}^{\infty }e^{2x}\,dx\)
\(\int_{0}^{\infty }e^{-x}\,dx\)
\(\int_{-\infty }^{-1}\dfrac{4}{x}\,dx\)
\(\int_{1}^{\infty }\dfrac{4}{x}\,dx\)
\(\int_{3}^{\infty }\dfrac{dx}{(x-1)^{4}} \)
\(\int_{-\infty }^{0}\dfrac{dx}{(x-1) ^{4}}\)
\(\int_{-\infty }^{\infty }\dfrac{dx}{x^{2}+4}\)
\(\int_{-\infty }^{\infty }\dfrac{dx}{x^{2}+1}\)
In Problems 25–32, determine whether each improper integral converges or diverges. If it converges, find its value.
\(\int_{0}^{1}\dfrac{dx}{x^{2}}\)
\(\int_{0}^{1}\dfrac{dx}{x^{3}}\)
\(\int_{0}^{1}\dfrac{dx}{x}\)
\(\int_{4}^{6}\dfrac{dx}{x-4}\)
\(\int_{0}^{4}\dfrac{dx}{\sqrt{4-x}}\)
\(\int_{1}^{5}\dfrac{x~dx}{\sqrt{5-x}}\)
\(\int_{-1}^{1}\dfrac{dx}{\sqrt[3]{x}}\)
\(\int_{0}^{3}\dfrac{dx}{(x-2)^{2}}\)
In Problems 33–62, determine whether each improper integral converges or diverges. If it converges, find its value.
\(\int_{0}^{\infty }\cos x\,dx\)
\(\int_{0}^{\infty }\sin (\pi x) \,dx\)
\(\int_{-\infty }^{0}e^{x}\,dx\)
\(\int_{-\infty }^{0}e^{-x}\,dx\)
\(\int_{0}^{\pi /2}\dfrac{x\,dx}{\sin x^{2}}\)
\(\int_{0}^{1}\dfrac{\ln x\,dx}{x}\)
\(\int_{0}^{1}\dfrac{dx}{1-x^{2}}\)
\(\int_{1}^{2}\dfrac{dx}{\sqrt{x^{2}-1}}\)
\(\int_{0}^{1}\dfrac{x\,dx}{(1-x^{2})^{2}}\)
\(\int_{0}^{2}\dfrac{dx}{(x-1)^{2}}\)
\(\int_{0}^{\pi /4}\tan (2x) \,dx\)
\(\int_{0}^{\pi /2}\csc x\,dx\)
\(\int_{0}^{\infty }\dfrac{x\,dx}{\sqrt{x+1}}\)
\(\int_{2}^{\infty }\dfrac{dx}{x\sqrt{x^{2}-1}}\)
\(\int_{-\infty }^{\infty }\dfrac{dx}{x^{2}+4x+5}\)
\(\int_{-\infty }^{\infty }\dfrac{dx}{e^{x}+e^{-x}}\)
\(\int_{-\infty }^{2}\dfrac{dx}{\sqrt{4-x}}\)
\(\int_{-\infty }^{1}\dfrac{x\,dx}{\sqrt{2-x}}\)
\(\int_{2}^{4}\dfrac{2x\,dx}{\sqrt[3]{x^{2}-4}}\)
\(\int_{0}^{\pi }\dfrac{1}{1-\cos x}\,dx\)
\(\int_{-1}^{1}\dfrac{1}{x^{3}}\,dx \)
\(\int_{0}^{2}\dfrac{dx}{x-1}\)
\(\int_{0}^{2}\dfrac{dx}{(x-1)^{1/3}}\)
\(\int_{-1}^{1}\dfrac{dx}{x^{5/3}}\)
\(\int_{1}^{2}\dfrac{dx}{(2-x)^{3/4}}\)
\(\int_{0}^{4}\dfrac{dx}{\sqrt{8x-x^{2}}}\)
\(\int_{a}^{3a}\dfrac{2x\,dx}{(x^{2}-a^{2})^{3/2}}\), \(a \gt 0\)
\(\int_{0}^{3}\dfrac{x\,dx}{(9-x^{2})^{3/2}}\)
\(\int_{0}^{\infty }xe^{-x^{2}}\,dx\)
\(\int_{0}^{\infty }e^{-x}\sin x\,dx\)
In Problems 63–70:
\(\int_{1}^{\infty }\dfrac{1}{\sqrt{x^{2}-1}}\,dx\)
\(\int_{2}^{\infty }\dfrac{2}{\sqrt{x^{2}-4}}\,dx\)
\(\int_{1}^{\infty }\dfrac{1+e^{-x}}{x}\,dx\)
\(\int_{1}^{\infty }\dfrac{3e^{-x}}{x}\,dx\)
\(\int_{1}^{\infty }\dfrac{\sin ^{2}x}{x^{2}}\,dx\)
\(\int_{1}^{\infty }\dfrac{\cos ^{2}x}{x^{2}}\,dx\)
\(\int_{1}^{\infty }\dfrac{dx}{(x+1) \sqrt{x}}\)
\(\int_{1}^{\infty }\dfrac{dx}{x\sqrt{1+x^{2}}}\)
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Applications and Extensions
Area Between Graphs Find the area, if it is defined, of the region enclosed by the graphs of \(y=\dfrac{1}{x+1}\) and \(y=\dfrac{1}{x+2}\) on the interval \([0,\infty )\). See the figure.
Area Between Graphs Find the area, if it is defined, under the graph of \(y=\dfrac{1}{1+x^{2}}\) to the right of \(x=0\). See the figure below.
Volume of a Solid of Revolution Find the volume, if it is defined, of the solid of revolution generated by revolving the region bounded by the graph of \(y=e^{-x}\) and the \(x\)-axis to the right of \(x=0\) about the \(x\)-axis. See the figure below.
Volume of a Solid of Revolution Find the volume, if it is defined, of the solid of revolution generated by revolving the region bounded by the graph of \(y=\dfrac{1}{\sqrt{x}}\) and the \(x\)-axis to the right of \(x=1\) about the \(x\)-axis. See the figure below.
Area Between Graphs Find the area, if it is defined, between the graph of \(y=\dfrac{8a^{3}}{x^{2}+4a^{2}}\), \(a>0\), and its horizontal asymptote.
Drug Reaction The rate of reaction \(r\) to a given dose of a drug at time \(t\) hours after administration is given by \(r(t)=t\,e^{-t^{2}}\) (measured in appropriate units).
Present Value of Money The present value \(PV\) of a capital asset that provides a perpetual stream of revenue that flows continuously at a rate of \(R(t)\) dollars per year is given by \[ PV= \int_{0}^{\infty }R(t)e^{-rt}\,dt \]
where \(r,\) expressed as a decimal, is the annual rate of interest compounded continuously.
Electrical Engineering In a problem in electrical theory, the integral \(\int_{0}^{\infty }Ri^{2}\,dt\) occurs, where the current \( i=Ie^{-Rt/L}\), \(t\) is time, and \(R\), \(I\), and \(L\) are positive constants. Find the integral.
Magnetic Potential The magnetic potential \(u\) at a point on the axis of a circular coil is given by \[ u=\dfrac{2\pi NIr}{10}\! \int_{x}^{\infty }\dfrac{dy}{(r^{2}+y^{2})^{3/2}} \]
where \(N\), \(I\), \(r\), and \(x\) are constants. Find the integral.
Electrical Engineering The field intensity \(F\) around a long (“infinite”) straight wire carrying electric current is given by the integral \[ F=\dfrac{rIm}{10}\int_{-\infty }^{\infty }\dfrac{dy}{(r^{2}+y^{2})^{3/2}} \]
where \(r,\) \(I\), and \(m\) are constants. Find the integral.
Work The force \(F\) of gravitational attraction between two point masses \(m\) and \(M\) that are \(r\) units apart is \(F=\dfrac{GmM}{r^{2}}\) , where \(G\) is the universal gravitational constant. Find the work done in moving the mass \(m\) along a straight-line path from \(r=1\) unit to \(r=\infty \).
For what numbers \(a\) does \(\int_{0}^{1}x^{a}\,dx\) converge?
In Problems 83–86, use integration by parts and perhaps L'Hôpital's Rule to find each improper integral.
\(\int_{0}^{\infty }xe^{-x}\,dx\)
\(\int_{0}^{1}x\ln x\,dx\)
\(\int_{0}^{\infty}e^{-x}\cos x\,dx\)
\(\int_{0}^{\infty }\tan^{-1}x\,dx \)
Show that \(\int_{0}^{\infty }\sin x\,dx\) and \(\int_{-\infty }^{0}\sin x\,dx\) each diverge, yet \(\lim\limits_{t\,\rightarrow \,\infty }\int_{-t}^{t}\sin x\, dx=0\).
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Find a function \(f\) for which \(\int_{0}^{\infty }f(x)\,dx\) and \(\int_{-\infty }^{0}f(x)\,dx\) each diverge, yet \(\lim\limits_{t\,\rightarrow \,\infty }\int_{-t}^{t}f(x)\,dx=1\).
Use the Comparison Test for Improper Integrals to show that \(\int_{0}^{\infty }\dfrac{1}{\sqrt{2+\sin x}}\,dx\) diverges.
Use the Comparison Test for Improper Integrals to show that \( \int_{2}^{\infty }\dfrac{\ln x}{\sqrt{x^{2}-1}}\,dx\) diverges.
If \(n\) is a positive integer, show that:
Show that \(\int_{e}^{\infty }\dfrac{\,dx}{x ( \ln x) ^{p}}\) converges if \(p \gt 1\) and diverges if \(p \le 1\).
Show that \(\int_{a}^{b}\dfrac{dx}{(x-a) ^{p}}\) converges if \(0 \lt p \lt 1\) and diverges if \(p \ge 1.\)
Show that \(\int_{a}^{b}\dfrac{dx}{(b-x)^{p}}\) converges if \(0 \lt p \lt 1\) and diverges if \(p \ge 1.\)
Refer to Problems 93 and 94. Discuss the convergence or divergence of the integrals if \(p \le 0.\) Support your explanation with an example.
Comparison Test for Improper Integrals Show that if two functions \(f\) and \(g\) are nonnegative and continuous on the interval \([ a,\infty ) \), and if \(f( x) \geq g( x) \) for all numbers \(x > c,\) where \(c \geq a,\) then
Laplace transforms are useful in solving a special class of differential equations. The Laplace transform L {f( x)} of a function \(f\) is defined as \[ L \{f( x)\} =\int_{0}^{\infty }e^{-sx} f(x) \,dx, x \ge 0, s \hbox{ a complex number } \]
In Problems 97–102, find the Laplace transform of each function.
\(f(x) =x\)
\(f(x) =\cos x\)
\(f(x) =\sin x\)
\(f(x) =e^{x} \)
\(f(x) =e^{ax}\)
\(f(x) =1\)
Challenge Problems
Find the arc length of \(y=\sqrt{x-x^{2}}-\sin ^{-1}\sqrt{x}\).
Find \(\int_{-\infty }^{a}e^{(x-e^{x})}\,dx.\)
Find \(\int_{-\infty }^{\infty }e^{(x-e^{x})}\,dx.\)
In Problems 107 and 108, use the following definition: A probability density function is a function \(f,\) whose domain is the set of all real numbers, with the following properties:
Uniform Density Function Show that the function \(f\) below is a probability density function. \[ f(x)=\left\{ \begin{array}{c@{\qquad}l@{\quad}rl} 0 & \hbox{if} & x& \lt a \\ \dfrac{1}{b-a} & \hbox{if} & a & \le x \le b, a \lt b \\ 0 & \hbox{if} & x& \gt b \end{array} \right. \]
Exponential Density Function Show that the function \(f\) is a probability density function for \(a>0\). \[ f(x)=\left\{ \begin{array}{l@{\qquad}l@{\quad}ll} \dfrac{1}{a}e^{-x/a} & \hbox{if}&x & \ge 0 \\ 0&\hbox{if}&x & \lt 0 \end{array} \right. \]
In Problems 109 and 110, use the following definition: The expected value or mean \(\mu\) associated with a probability density function \(f\) is defined by \[ \mu =\int_{-\infty }^{\infty }x f(x)\,dx \]
The expected value can be thought of as a weighted average of its various probabilities.
Find the expected value \(\mu \) of the uniform density function \(f\) given in Problem 107.
Find the expected value \(\mu \) of the exponential probability density function \(f\) defined in Problem 108.
In Problems 111 and 112, use the following definitions: The variance \(\sigma ^{2}\) of a probability density function \(f\) is defined as \[ \sigma ^{2}=\int_{-\infty }^{\infty }(x-\mu )^{2} f(x)\,dx \] The variance is the average of the squared deviation from the mean. The standard deviation \(\sigma \) of a probability density function \(f\) is the square root of its variance \(\sigma ^{2}\).
Find the variance \(\sigma ^{2}\) and standard deviation \( \sigma \) of the uniform density function defined in Problem 107.
Find the variance \(\sigma ^{2}\) and standard deviation \( \sigma \) of the exponential density function defined in Problem 108.