Math 2565, Spring 2020

Improper Integrals

January 21, 2020

Sean Fitzpatrick
University of Lethbridge

Warm-Up

Trig substitution:

\begin{equation*} \int\frac{x}{(x^2+9)^{3/2}}\,dx \end{equation*}
\begin{equation*} \int\frac{\sqrt{x^2-1}}{x}\,dx \end{equation*}

Partial fractions:

\begin{equation*} \int\frac{x^3}{x^2-4}\,dx \end{equation*}
\begin{equation*} \int\frac{2x^2+x+1}{(x+1)(x^2+9)}\,dx \end{equation*}

Quiz

  1. For which values of \(p\) does the integral

    \begin{equation*} \int_1^\infty\frac{1}{x^p}\,dx \end{equation*}
    converge?

  2. Which of the following integrals would be considered improper?

    1. \(\displaystyle \int_1^\infty\frac{1}{x^2+1}\,dx\)
    2. \(\displaystyle \int_0^1\frac{1}{\sqrt{x}}\,dx\)

    3. An integral that uses the same fork for its main course and dessert.

    4. All of the above.

Improper integrals

Although we can evaluate many improper integrals, the main question is not “What is the value?”, but “Does it converge?”

Improper integrals are potentially infinite. An improper integral converges if it has a finite value.

Definition:

For a function \(f(x)\) continuous on \([a,\infty)\text{,}\) we set

\begin{equation*} \int_a^\infty f(x)\,dx = \lim_{b\to \infty}\int_a^b f(x)\,dx\text{,} \end{equation*}
provided this limit exists.

If \(f(x)\) is unbounded at \(x=c\text{,}\) we set

\begin{equation*} \int_a^c f(x)\,dx = \lim_{b\to c^-}\int_a^b f(x)\,dx \end{equation*}
and
\begin{equation*} \int_c^b f(x)\,dx = \lim_{a\to c^+}\int_a^bf(x)\,dx\text{.} \end{equation*}

Examples: unbounded domain

  1. \(\di \int_1^\infty \frac{1}{x^4}\,dx\)

  2. \(\di \int_3^\infty \frac{x}{x^2-4}\,dx\)

  3. \(\di \int_{-\infty}^\infty \frac{1}{x^2+16}\,dx\)

  4. \(\di \int_0^\infty e^{-5x}\,dx\)

  5. \(\di \int_5^\infty\frac{1}{x-4}\,dx\)

The \(p\)-test

Theorem:

The improper integral \(\di \int_1^\infty\frac{1}{x^p}\,dx\) converges if \(p\gt 1\text{,}\) and diverges if \(p\leq 1\text{.}\)

Examples: unbounded range

  1. \(\di \int_0^1\ln(x)\,dx\)

  2. \(\di \int_0^1 \frac{1}{\sqrt[3]{x}}\,dx\)

  3. \(\di \int_0^1\frac{1}{x^p}\,dx\) (For what \(p\) does it converge?)

Comparison

Premise is simple:

  • If \(f(x)\leq g(x)\) and \(\int_a^\infty g(x)\,dx\) converges, so does \(\int_a^\infty f(x)\,dx\text{.}\)

    If \(f(x)\geq g(x)\) and \(\int_a^\infty g(x)\,dx\) diverges, so does \(\int_a^\infty f(x)\,dx\text{.}\)

The hard part is setting up the comparison.

Examples

Use comparison to decide on convergence or divergence of the improper integral:

  1. \(\di \int_2^\infty\frac{1}{x^2+x}\,dx\)

  2. \(\di \int_1^\infty\frac{1}{\sqrt{x^2-1}}\,dx\)

  3. \(\di \int_2^\infty\frac{1}{\sqrt{x^2+1}}\,dx\)

What about \(\di \int_3^\infty\frac{1}{x\ln(x)}\,dx\text{?}\)

Limit comparison

Useful when direct comparison is hard to set up.

Theorem:

Suppose \(\di \lim_{x\to\infty}\frac{f(x)}{g(x)}=L\text{,}\) where \(0\lt L\lt \infty\text{.}\) Then the integrals

\begin{equation*} \int_a^\infty f(x)\,dx \quad\quad \int_a^\infty g(x)\,dx \end{equation*}
both converge, or both diverge.

Example:

Determine the convergence of

\begin{equation*} \int_10^\infty\frac{1}{x^3-2x^2+4x+3}\,dx \quad \text{ and } \quad \int_4^\infty \frac{x}{x^2+\sin(x)}\,dx\text{.} \end{equation*}

If time...

What can we say about \(\di \int_4^\infty\frac{x^2}{e^x}\,dx\text{?}\)