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17Calculus - Improper Integrals on Infinite Intervals

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

This page covers how to evaluate improper integrals on infinite intervals. As you learned on the previous page, basic definite integration can be done only on intervals that are continuous and finite. First, we discuss how to handle the case when the interval is not finite, i.e. one or both limits of integration are infinite.

Case 1 - Infinite Interval

Case 1: Infinite Integral at One Endpoint - Problem and Example

\(\displaystyle{ \int_{a}^{\infty}{f(x) ~ dx} }\)

\(a\) is finite

\(f(x)\) is continuous on the interval \([a,\infty)\)

\(\displaystyle{ \int_{1}^{\infty}{\frac{1}{x^2} ~ dx} }\)

\(\displaystyle{ \int_{-\infty}^{b}{g(x) ~ dx} }\)

\(b\) is finite

\(g(x)\) is continuous on the interval \( (-\infty,b] \)

\(\displaystyle{ \int_{-\infty}^{-1}{\frac{1}{x^2} ~ dx} }\)

Special Improper Integral

\(\displaystyle{ \int_{1}^{\infty}{\frac{dx}{x^p}} = \left\{ \begin{array}{llr} \displaystyle{\frac{1}{p-1}} & & p \gt 1 \\ \text{diverges} & & p \leq 1 \end{array} \right. }\)

Notes
1. We chose the function \(1/x^2\) to demonstrate what we meant about being continuous on the interval of integration. Notice that as long as we stay away from \(x=0\), the function \(f(x)=1/x^2\) is continuous.
2. Also notice that we have a problem at only one endpoint in each integral. This is important, since we can handle only one problem at a time.
3. Using the techniques we know so far, we can't evaluate these integrals because it is required that the function be continuous and the definite integral can be evaluated only on a finite interval. So what do we do?

This is the case discussed in this video. He looks at the same function \(1/x^2\) and shows how to evaluate it with limits. He uses a graph to explain really well what is going on with this type of improper integral.

PatrickJMT - Improper Integral - Basic Idea and Example [6min-22secs]

video by PatrickJMT

So, to evaluate the integrals in the above table, we need these conditions.

Under These Conditions

\(a\), \(b\) and \(t\) are finite

\(f(x)\) is continuous on the interval \([a,\infty)\)

\(g(x)\) is continuous on the interval \( (-\infty,b] \)

We Can Write These Equations

\(\displaystyle{ \int_{a}^{\infty}{f(x) ~ dx} = }\) \(\displaystyle{\lim_{t \to \infty}{ \int_{a}^{t}{f(x) ~ dx} } }\)

\(\displaystyle{ \int_{-\infty}^{b}{g(x) ~ dx} = }\) \(\displaystyle{\lim_{t \to -\infty}{ \int_{t}^{b}{g(x) ~ dx} } }\)

Notice what we have done. We have replaced the infinity with a finite variable. This allows us to evaluate the integral. Once the integral is evaluated, we can then take the limit of the result. So, in essence, we have moved the 'problem' outside of the integral.

This is an important starting point since the upcoming cases rely on your understanding of this first case. Take a few minutes to work these practice problems before you go on.

Practice

Unless otherwise instructed, evaluate these integrals using proper notation. Give your answers in exact, simplified and factored form.

Basic

\(\displaystyle{\int_{0}^{\infty}{\frac{dx}{(x+1)^3}}}\)

Problem Statement

Evaluate \(\displaystyle{\int_{0}^{\infty}{\frac{dx}{(x+1)^3}}}\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Final Answer

\(1/2\)

Problem Statement

Evaluate \(\displaystyle{\int_{0}^{\infty}{\frac{dx}{(x+1)^3}}}\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

Looking at the limits of integration, we have the upper limit at infinity. So we write the integral with a limit before integrating.

\(\displaystyle{ \lim_{b\to\infty}{ \int_{0}^{b}{ \frac{dx}{(x+1)^3} } } }\)

\(\displaystyle{ \lim_{b\to\infty}{ \int_{0}^{b}{ (x+1)^{-3} ~dx } } }\)

\(\displaystyle{ \lim_{b\to\infty}{ \left. \frac{(x+1)^{-2}}{-2} \right|_0^b } }\)

\(\displaystyle{ \lim_{b\to\infty}{ \left[ \frac{-1}{2(b+1)^2} - \frac{-1}{2(1)^2} \right] } = 0 + \frac{1}{2} = \frac{1}{2} }\)

Final Answer

\(1/2\)

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\(\displaystyle{ \int_{1}^{\infty}{ 2^{-x} ~dx }}\)

Problem Statement

Evaluate \(\displaystyle{ \int_{1}^{\infty}{ 2^{-x} ~dx }}\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Final Answer

\(\displaystyle{ \frac{1}{\ln 4} }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{1}^{\infty}{ 2^{-x} ~dx }}\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

First, let's write the integrand in terms of \(e\) so that integration is easier.

\(y=2^{-x}\)

\(\ln y = \ln 2^{-x}\)

\(\ln y = -x \ln 2\)

\( e^{\ln y} = e^{-x\ln 2} \)

\( y = e^{-x\ln 2} \)

So we will replace the integrand \(2^{-x}\) with \( e^{-x\ln 2} \).
Rewrite the integral with a limit and integrate.

\(\displaystyle{ \lim_{b\to\infty}{ \int_{1}^{b}{ e^{-x\ln 2} ~dx } } }\)

\(\displaystyle{ \lim_{b\to\infty}{ \left. \frac{e^{-x\ln2}}{-\ln2} \right|_1^b } }\)

\(\displaystyle{ \lim_{b\to\infty}{ \left[ \frac{-1}{e^{b\ln2}\ln 2} - \frac{-1}{e^{\ln2}\ln 2} \right] } }\)

\(\displaystyle{ 0 + \frac{1}{2\ln 2} = \frac{1}{\ln 4} }\)

Final Answer

\(\displaystyle{ \frac{1}{\ln 4} }\)

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\(\displaystyle{ \int_{1}^{\infty}{ x^2~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{1}^{\infty}{ x^2~dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1464 video solution

video by MIP4U

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\(\displaystyle{ \int_{1}^{\infty}{ \frac{1}{x} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{1}^{\infty}{ \frac{1}{x} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

This problem is solved by two different instructors in these two videos.

The Organic Chemistry Tutor - 1465 video solution

MIP4U - 1465 video solution

video by MIP4U

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\(\displaystyle{ \int_{1}^{\infty}{ \frac{1}{x^2} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{1}^{\infty}{ \frac{1}{x^2} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

The Organic Chemistry Tutor - 1287 video solution

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\(\displaystyle{ \int_{2}^{\infty}{ \frac{dx}{x^5} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{2}^{\infty}{ \frac{dx}{x^5} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

Make sure you understand that you do not plug in the infinite limit as if it was a number after evaluating the integral, like she does in this video. What is actually happening is that you are taking the limit. Before starting any work, the initial integral should be written
\(\displaystyle{ \int_{2}^{\infty}{ \frac{dx}{x^5} dx } = \lim_{b\to\infty}{\int_{2}^{b}{ \frac{dx}{x^5} dx } } }\)

Krista King Math - 994 video solution

video by Krista King Math

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\(\displaystyle{ \int_{-\infty}^{0}{ e^{0.1x} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^{0}{ e^{0.1x} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

Make sure you understand that you do not plug in the infinite limit as if it was number after evaluating the integral, like she does in this video. What is actually happening is that you are taking the limit. Before starting any work, the initial integral should be written
\(\displaystyle{ \int_{-\infty}^{0}{ e^{0.1x} dx } = \lim_{a\to-\infty}{\int_{a}^{0}{ e^{0.1x } dx } } }\)

Krista King Math - 995 video solution

video by Krista King Math

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\(\displaystyle{ \int_{1}^{\infty}{ 2x^{-2} ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{1}^{\infty}{ 2x^{-2} ~dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1463 video solution

video by MIP4U

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\(\displaystyle{ \int_{0}^{\infty}{ 4e^{-2x} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{0}^{\infty}{ 4e^{-2x} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1466 video solution

video by MIP4U

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\(\displaystyle{ \int_{0}^{\infty}{ 6e^{-3x} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{0}^{\infty}{ 6e^{-3x} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1469 video solution

video by MIP4U

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\(\displaystyle{ \int_{0}^{\infty}{ \frac{1}{2x+5} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{0}^{\infty}{ \frac{1}{2x+5} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1471 video solution

video by MIP4U

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\(\displaystyle{ \int_{1}^{\infty}{ \frac{1}{(3x+1)^2} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{1}^{\infty}{ \frac{1}{(3x+1)^2} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

After initially solving this problem, he reminds us how to evaluate a limit at infinity. This is a good review.

The Organic Chemistry Tutor - 1288 video solution

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\(\displaystyle{ \int_{-\infty}^{-3}{ \frac{2}{x^2} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^{-3}{ \frac{2}{x^2} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1472 video solution

video by MIP4U

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\(\displaystyle{ \int_{0}^{\infty}{ 0.5e^xdx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{0}^{\infty}{ 0.5e^xdx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1475 video solution

video by MIP4U

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\(\displaystyle{ \int_{1}^{\infty}{ \frac{8}{x^4} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{1}^{\infty}{ \frac{8}{x^4} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1476 video solution

video by MIP4U

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\(\displaystyle{ \int_{-\infty}^{0}{ \frac{dx}{3-4x} } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^{0}{ \frac{dx}{3-4x} } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

Krista King Math - 1820 video solution

video by Krista King Math

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Intermediate

\(\displaystyle{ \int_{0}^{\infty}{ x e^{-3x} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{0}^{\infty}{ x e^{-3x} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

This practice problem is solved in two consecutive videos.
Her notation here is not good and she makes a major jump (without explaining something) when evaluating one of the limits.
Her notation mistake is that she didn't write the integral as a limit. This may seem trivial but it isn't since notation will often be what will trip you up. So she should have started the problem by rewriting the integral as \(\displaystyle{\int_{0}^{\infty}{x e^{-3x} dx} = \lim_{b \to \infty}{ \int_{0}^{b}{x e^{-3x} dx}}}\) then worked the inside integral and evaluated the limit at the end.
Secondly, she evaluates the limit of one of the terms and said it was zero when actually it is indeterminate. The term was \(\displaystyle{\lim_{b \to \infty}{ \frac{-be^{-3b}}{3}}}\).
Since she didn't write out the limit, this may be hard to see but it is the first term when she plugged in \( \infty \). The term she got was \(\displaystyle{\frac{-\infty e^{-3\infty}}{3}}\) which can be rewritten as \(\displaystyle{\frac{-\infty }{3e^{3\infty}} = \frac{-\infty}{\infty}}\) and is clearly indeterminate. To evaluate the limit, we need to use L'Hopitals rule as follows
\(\displaystyle{ \lim_{b \to \infty}{ \frac{-be^{-3b}}{3} } = }\) \(\displaystyle{ \lim_{b \to \infty}{ \frac{-b}{3e^{3b}} } = }\) \(\displaystyle{ \lim_{b \to \infty}{ \frac{-1}{9e^{3b}} } = \frac{-1}{\infty} = 0 }\)

So, that is where the zero comes from in that term. She either got lucky or she just didn't show the steps, neither of which should be taken for granted. Other than these mistakes, this is a good example.

Krista King Math - 993 video solution

video by Krista King Math

Krista King Math - 993 video solution

video by Krista King Math

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\(\displaystyle{ \int_{0}^{\infty}{ se^{-5s} ~ds } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{0}^{\infty}{ se^{-5s} ~ds } }\) giving your answer in exact, simplified, factored form. Make sure to use correct notation in every step of your work.

Solution

PatrickJMT - 1460 video solution

video by PatrickJMT

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\(\displaystyle{ \int_{-\infty}^{2}{ \frac{3}{x^2+1} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^{2}{ \frac{3}{x^2+1} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1473 video solution

video by MIP4U

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\(\displaystyle{ \int_{0}^{\infty}{ \cos(x) ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{0}^{\infty}{ \cos(x) ~dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIT OCW - 1478 video solution

video by MIT OCW

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\(\displaystyle{ \int_{-\infty}^0{ x e^x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^0{ x e^x ~dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Final Answer

\( -1 \)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^0{ x e^x ~dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

This problem is solved by two different instructors in these two videos.

The Organic Chemistry Tutor - 2223 video solution

Michel vanBiezen - 2223 video solution

video by Michel vanBiezen

Final Answer

\( -1 \)

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So, you should now be comfortable evaluating integrals where the integrand is continuous on the entire interval of integration and with one of the endpoints going to infinity or negative infinity. But what do you do if both limits of integration contain infinity, i.e. the lower limit is negative infinity AND the upper limit is infinity. The trick is to break the integral at a convenient point and then you have two integrals. For example, if we have \(\displaystyle{ \int_{-\infty}^{\infty}{ x ~ dx} }\), we can break the integral at any point \(a\) to get integrals.
\(\displaystyle{ \int_{-\infty}^{\infty}{ x ~ dx} = }\) \(\displaystyle{ \int_{-\infty}^{0}{ x } + }\) \(\displaystyle{ \int_{0}^{\infty}{ x ~ dx} }\)
We chose zero in the above example but you can choose any point or even use \(a\). Then we use the technique we learned above to evaluate each integral, adding the results together to get our final answer.

Here are some practice problems to try.

Practice

Unless otherwise instructed, evaluate these integrals using proper notation. Give your answers in exact, simplified and factored form.

\(\displaystyle{ \int_{-\infty}^{\infty}{ -3x^4 ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^{\infty}{ -3x^4 ~dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1470 video solution

video by MIP4U

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\(\displaystyle{ \int_{-\infty}^{\infty}{ x^2 e^{-x^3} ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^{\infty}{ x^2 e^{-x^3} ~dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

PatrickJMT - 1216 video solution

video by PatrickJMT

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\(\displaystyle{ \int_{-\infty}^{\infty}{ \frac{1}{1+x^2} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^{\infty}{ \frac{1}{1+x^2} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

MIP4U - 1477 video solution

video by MIP4U

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\(\displaystyle{ \int_{-\infty}^{\infty}{ \frac{4}{16+x^2} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^{\infty}{ \frac{4}{16+x^2} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

The Organic Chemistry Tutor - 1289 video solution

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\(\displaystyle{ \int_{-\infty}^{\infty}{ \frac{x^2}{9+x^6} dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int_{-\infty}^{\infty}{ \frac{x^2}{9+x^6} dx } }\) giving your answer in exact, simplified and factored form. Make sure to use correct notation in every step of your work.

Solution

Krista King Math - 1821 video solution

video by Krista King Math

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Practice Instructions

Unless otherwise instructed, evaluate these integrals using proper notation. Give your answers in exact, simplified and factored form.

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