# Improper Integrals with Infinite Intervals

 

Define and calculate improper integrals with infinite limits of integration through examples with detailed solutions and explanations. Also more exercises with solutions are included.

## Definition of Improper Integrals with Infinite Intervals [1] [2]

Consider 3 types of integral with infinite interval(s)

### 1 - The Upper Limit of the Integral is Infinite

An integral of the form $\displaystyle \int_a^{\infty} f(x) \; dx$ exists and is convergent if $\displaystyle \int_a^{b} f(x) \; dx \quad \text{exists for all} \quad b \ge a$ and the limit $\lim_{b \to \infty} \displaystyle \int_a^{b} f(x) \; dx$ exists and is finite.
We then write
$\boxed{ \displaystyle \int_a^{\infty} f(x) \; dx = \lim_{b \to \infty} \displaystyle \int_a^{b} f(x) \; dx }$

### 2 - The Lower Limit of the Integral is Infinite

An integral of the form $\displaystyle \int_{-\infty}^a f(x) \; dx$ exists and is convergent if $\displaystyle \int_b^{a} f(x) \; dx \quad \text{exists for all} \quad b \le a$ and the limit $\lim_{b \to -\infty} \displaystyle \int_b^{a} f(x) \; dx$ exists and is finite.
We then write
$\boxed{ \displaystyle \int_{-\infty}^a f(x) \; dx = \lim_{b \to -\infty} \displaystyle \int_b^{a} f(x) }$

### 3 - Both Limits of the Integral are Infinite

An integral of the form $\displaystyle \int_{-\infty}^{\infty} f(x) \; dx$ exists and is convergent if both $\displaystyle \int_{-\infty}^a f(x) \; dx$ and $\displaystyle \int_a^{\infty} f(x) \; dx$ are convergent
We then write
$\boxed{ \displaystyle \int_{-\infty}^{\infty} f(x) \; dx = \displaystyle \int_{-\infty}^a f(x) \; dx + \displaystyle \int_a^{\infty} f(x) \; dx }$
Note that the interval of integration is split at $x = a$ and hence we end up with the sum of two integrals with each integral having one infinite limit only.

## Examples and Their Solutions

Example 1
Evaluate the integral given below if possible. $\displaystyle \int_1^{\infty}\; x \; dx$

Example 2
Is the integral below convergent or divergent? Evaluate it if it is convergent. $\displaystyle \int_1^{\infty}\; \dfrac{1}{x^2} \; dx$

Example 3
Evaluate the integral given below if possible. $\displaystyle \int_{-\infty}^0\; \dfrac{1}{\sqrt{2-3x}} \; dx$

Example 4
Evaluate the integral given below if possible. $\displaystyle \int_{-\infty}^{\infty}\; x^5 e^{-x^6} \; dx$

Example 5
For what values of $p$ is the integral $\displaystyle \int_{e}^{\infty}\; \dfrac{1}{x\:\left(\ln\:x\right)^{p+1}} \; dx$ convergent? Find its value in terms of $p$ .

## Exercises

Evaluate each of the following integrals if possible.

1. $\displaystyle \int _1^{\infty } \; \dfrac{1}{\left(4x+1\right)^2} \; dx$

2. $\displaystyle \int _{-\infty }^{\infty } \; \dfrac{x}{1+x^2} \; dx$

3. $\displaystyle \int _{-\infty }^{-1} \; e^{2x} \; dx$

4. $\displaystyle \int _{-\infty }^{\infty }\dfrac{7\:x^2}{3+x^6} \; dx$

5. $\displaystyle \int _0^{\infty }\dfrac{e^{2x}}{e^{4x}+3} \; dx$