Improper integrals

Section 1.24 Improper integrals

Definition 1.24.1. Improper integral of type I: infinite intervals.

Below we define definite integrals over infinite intervals. These are called improper integrals of type I, or integrals over infinite intervals.

Interval \([a,\infty)\).

Let \(f\) be continuous on the interval \([a,\infty)\text{.}\) The integral of \(f\) over \([a,\infty)\), denoted \(\int_a^\infty f(x)\, dx\text{,}\) is said to converge if the limit

\begin{equation} \lim_{R\to\infty}\int_a^R f(x)\, dx\tag{1.24.1} \end{equation}

exists, in which case we define

\begin{equation} \int_a^\infty f(x)\, dx=\lim_{R\to\infty}\int_a^R f(x)\, dx\text{.}\tag{1.24.2} \end{equation}

If the limit (1.24.1) does not exist, we say the improper integral diverges.
Interval \((-\infty,a]\).

Let \(f\) be continuous on the interval \((-\infty,a]\text{.}\) The integral of \(f\) over \((-\infty,a]\), denoted \(\int_{-\infty}^a f(x)\, dx\text{,}\) is said to converge if the limit

\begin{equation} \lim_{R\to\infty}\int_{-R}^a f(x)\, dx\tag{1.24.3} \end{equation}

exists, in which case we define

\begin{equation} \int_{-\infty}^a f(x)\, dx=\lim_{R\to\infty}\int_{-R}^a f(x)\, dx\text{.}\tag{1.24.4} \end{equation}

If the limit (1.24.3) does not exist, we say the improper integral diverges.
Real line.

Let \(f\) be continuous on the real line \(\R=(-\infty,\infty)\text{.}\) The integral of \(f\) over \((-\infty,\infty)\), denoted \(\int_{-\infty}^{\infty}f(x)\, dx\text{,}\) is said to converge if both of the half-infinite integrals \(\int_{-\infty}^a f(x)\, dx\) and \(\int_a^{\infty}f(x)\, dx\) converge for some real number \(a\text{.}\) In this case we define

\begin{equation} \int_{-\infty}^\infty f(x)\, dx=\int_{-\infty}^a f(x)\, dx+\int_a^{\infty}f(x)\, dx\text{.}\tag{1.24.5} \end{equation}

If either (or both) of the half-infinite integrals diverge, we say that the integral of \(f\) over \((-\infty, \infty)\) diverges.

Procedure 1.24.2. Improper integral: type I.

To evaluate an improper integral of the form \(\int_a^\infty f(x)\, dx\) or \(\int_{-\infty}^af(x)\, dx\text{,}\) proceed as follows.

Compute the relevant definite integral \(I_R\text{,}\) that is \(I_R=\int_a^Rf(x)\, dx\) or \(I_R=\int_{-R}^af(x)\, dx\text{,}\) as an expression in terms of \(R\text{.}\)
Investigate the relevant limit

\begin{align*} \lim_{R\to\infty}I_R\amp =\int_a^R f(x)\, dx \amp \\ \lim_{R\to\infty}I_R\amp =\lim_{R\to\infty}\int_{-R}^af(x)\, dx \amp \text{.} \end{align*}
If the relevant limit does not exist, conclude that the improper integral diverges. Otherwise, conclude that the improper integral is equal to the relevant limit: i.e.,

\begin{align*} \int_a^\infty f(x)\, dx\amp =\lim_{R\to\infty}\int_a^R f(x)\, dx \\ \int_{-\infty}^a f(x)\, dx\amp =\lim_{R\to\infty}\int_{-R}^af(x)\, dx \amp \text{.} \end{align*}

To evaluate a definite integral of the form \(\int_{-\infty}^{\infty}f(x)\, dx\text{,}\) pick any real number \(a\text{,}\) and apply the proceeding to both \(\int_a^\infty f(x)\, dx\) and \(\int_{-\infty}^af(x)\, dx\text{.}\) The improper integral over \((-\infty, \infty)\) exists if and only if both of these half-infinite integrals exist, in which case it is equal to their sum.

Example 1.24.3. Type I: half-infinite.

Evaluate \(\displaystyle\int_{-2}^{\infty}e^{-x}\, dx\text{.}\)

Solution.

Example 1.24.4. Type I: \(p\)-test.

Evaluate \(\displaystyle\int_{2}^\infty \frac{1}{x^p}\, dx\) for \(p> 0\text{.}\)

Solution.

Theorem 1.24.5. Improper integrals: \(p\)-test.

Let \(a\) and \(p\) be any positive numbers. We have

\begin{equation} \int_a^\infty \frac{1}{x^p}\, dx=\begin{cases} \frac{1}{(p-1)a^{p-1}} \amp \text{if } p> 1 \\ \infty \amp \text{if } p\leq 1. \end{cases} \text{.}\tag{1.24.6} \end{equation}

Example 1.24.6. Type I: half-infinite.

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

Solution.

Definition 1.24.7. Improper integrals of type II: discontinuities.

Assume \(f\) is continuous on the interval \(I=[a,b]\text{,}\) except possibly at one point.

Assume \(f\) is not continuous at \(x=a\text{.}\) The integral of \(f\) over \([a,b]\), denoted \(\int_a^b f(x)\, dx\text{,}\) is said to converge if the limit

\begin{equation} \lim_{t\to a^+}\int_t^b f(x)\, dx\text{,}\tag{1.24.7} \end{equation}

exists, in which case we define

\begin{equation} \int_a^b f(x)\, dx=\lim_{t\to a^+}\int_t^b f(x)\, dx\text{.}\tag{1.24.8} \end{equation}

If the limit (1.24.7) does not exist, we say the improper integral diverges.
Assume \(f\) is not continuous at \(x=b\text{.}\) The integral of \(f\) over \([a,b]\), denoted \(\int_a^b f(x)\, dx\text{,}\) is said to converge if the limit

\begin{equation} \lim_{t\to b^-}\int_a^t f(x)\, dx\text{,}\tag{1.24.9} \end{equation}

exists, in which case we define

\begin{equation} \int_a^b f(x)\, dx=\lim_{t\to b^-}\int_a^t f(x)\, dx\text{.}\tag{1.24.10} \end{equation}

If the limit (1.24.7) does not exist, we say the improper integral diverges.
Assume \(f\) is not continuous at \(c\in (a,b)\text{.}\) The integral of \(f\) over \([a,b]\), denoted \(\int_a^bf(x)\, dx\text{,}\) is said to converge if both improper integrals \(\int_a^cf(x)\, dx\) and \(\int_c^b f(x)\, dx\) converge, in which case we define

\begin{equation} \int_a^bf(x)\, dx=\int_a^c f(x)\, dx+ \int_c^b f(x)\, dx,\text{.}\tag{1.24.11} \end{equation}

Otherwise, we say that the integral over the entire interval diverges.

Procedure 1.24.8. Improper integral: type II.

To evaluate an improper integral of the form \(\int_a^b f(x)\, dx\text{,}\) where \(f\) fails to be continuous at at most one of the endpoints of \([a,b]\text{,}\) proceed as follows.

Compute the relevant definite integral, that is \(\int_t^b f(x)\, dx\) or \(\int_a^t f(x)\, dx\text{,}\) as an expression in terms of \(t\text{.}\)
Investigate the relevant limit

\begin{align*} \lim_{t\to a^+}\int_t^b f(x)\, dx \amp \\ \lim_{t\to b^-}\int_{a}^tf(x)\, dx \amp \text{.} \end{align*}
If the relevant limit does not exist, conclude that the improper integral diverges. Otherwise, conclude that the improper integral is equal to the relevant limit: i.e.,

\begin{align*} \int_a^b f(x)\, dx\amp =\lim_{t\to a^+}\int_t^b f(x)\, dx \\ \int_{a}^b f(x)\, dx\amp =\lim_{t\to b^-}\int_{a}^tf(x)\, dx \amp \text{.} \end{align*}

To evaluate a definite integral of the form \(\int_{a}^{b}f(x)\, dx\text{,}\) where \(f\) is continuous everywhere except at \(x=c\in (a,b)\text{,}\) apply the proceeding to both \(\int_a^c f(x)\, dx\) and \(\int_{c}^bf(x)\, dx\text{.}\) The improper integral over the \([a,b]\) exists if and only if both of these smaller interval integrals exist, in which case it is equal to their sum.

Example 1.24.9. Improper: type II.

Evaluate \(\displaystyle\int_{0}^2\frac{1}{x-1}\, dx\text{.}\)

Solution.

Example 1.24.10. Improper: type II.

Evaluate \(\displaystyle\int_0^{1}\ln x\, dx\text{.}\)

Solution.

Example 1.24.11. Improper: type II.

Evaluate \(\displaystyle\int_1^{4}\frac{x}{\sqrt[3]{x^2-4}}\, dx\text{.}\)

Solution.

Definition 1.24.12. Area interpretation of improper integrals.

Let \(f\) be defined on an interval \(I\) for which the corresponding integral is improper, and let \(\mathcal{R}\) be the (potentially unbounded) region between the graph of \(f\) and the \(x\)-axis over the interval \(I\text{.}\)

We define the area (or total area) of \(\mathcal{R}\) to be the integral of \(\lvert f\rvert\) over \(I\text{,}\) assuming this integral converges.
We define the signed area of \(\mathcal{R}\) to be the integral of \(f\) over \(I\text{,}\) assuming this interval converges.

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