# 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)

#### 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.