Chapter 49: Improper Integrals

We have dealt with definite integrals like:

$\int_{2}^{5}x e^{x^{2}}dx$

Where the domain and range are both finite.

Definite Integral - is Improper if- the domain or range is unbounded.

Example

$-\int_{1}^{\infty}e^{-x}dx$

$=\lim_{t\rightarrow \infty}-\int_{1}^{t}e^{-x}dx$

$=\lim_{t\rightarrow \infty}[-e^{-x}]_{1}^{t}$

$=\lim_{t\rightarrow \infty}[-e^{-x}]-[-e^{-1}]]$ (goes to 0 as $t \rightarrow \infty$)

$=e^{-1}$

$\therefore$ The region is unbounded but it has finite total area (and infinite perimeter)

Example

Discuss

Find $\int_{0}^{CS} \frac{1}{1+x^{2}}dx$

• Method- to find $\int_{C}^{CS}f\left ( x \right )dx$ $= lim \int_{t}^{C}f\left ( x \right )dx$

$t\rightarrow CS$

or $\int_{-CS}^{C}f\left ( x \right )dx$ $= lim \int_{C}^{t}f\left ( x \right )dx$

$t\rightarrow CS$

Example

$\int_{-CS}^{CS} \frac{e^{x}}{e^{2x+1}}dx$

Spit up the integral (reduce to known cases)

$\int_{0}^{CS} \frac{e^{x}}{e^{2x+1}}dx+\int_{-CS}^{0}\frac{e^{x}}{e^{2x+1}}dx$

Apply substitution $u= e^{x}\Rightarrow du=e^{x}$

An improper integral is Convergent if,

It is Divergent otherwise,

Example

$\int_{1}^{CS}\frac{1}{x}dx$

Thus, the integral is divergent.

Question

When is $\int_{1}^{CS}\frac{1}{x}dx$ convergent?

We need $p> 1$ for this limit to be finite

We get-

Fact: If $p> 1$ , then $\int_{1}^{CS}\frac{1}{x}dx= \frac{1}{p-1}$

Unbounded Range

Fact: if $f(x)$ has a unique asymptote

Thus, the total area is finite.