# Solving a very strange improper integral

• Susanne217
In summary, the conversation is discussing an improper integral and how to obtain a specific result. The attempt at a solution involves splitting the integral into two parts and using the relationships involving arctan. However, the result obtained is not equal to the desired result. The questioner is unsure if x is a constant term in the problem.
Susanne217

## Homework Statement

I am getting fooled by the this improper integral

$$\int_0^{\infty}\frac{cos(x)+sin(x)}{1+v^2}dv = \pi \cdot e^{-x}$$

How the devil do I go about getting that result?

## The Attempt at a Solution

I end up getting the sum of the two integrals

$$\int_0^{1}\frac{cos(x)}{1+v^2}dv + \int_1^{\infty}\frac{sin(x)}{1+v^2}dv$$ but how do I proceed from there? Hints please :)

is x a constant term here?

Since you are integrating with respect to v, this is just
$$(cos(x)+ sin(x))\int_0^\infty \frac{dv}{1+ v^2}$$

And you should know that
$$\int\frac{dv}{1+ v^2}= arctan(v)+ C$$.

But that is NOT equal to $\pi e^{-x}$. You must have copied the problem incorrectly or left out some relationship between x and v.

## What is an improper integral?

An improper integral is an integral where either the upper or lower limit of integration is infinite, or the integrand is undefined at one or more points within the limits of integration. These types of integrals require special techniques to evaluate.

## How do you solve an improper integral?

To solve an improper integral, you must first determine if it is convergent or divergent. If it is convergent, you can use a variety of techniques such as integration by parts, substitution, or partial fractions to evaluate the integral. If it is divergent, you can use limits to determine its behavior.

## What are some common strategies for solving improper integrals?

Some common strategies for solving improper integrals include using symmetry, breaking up the integral into smaller parts, and using trigonometric identities. Additionally, using a substitution to change the form of the integral can also be helpful.

## What challenges may arise when solving a very strange improper integral?

Some challenges that may arise when solving a very strange improper integral include determining the appropriate limits of integration, dealing with undefined or discontinuous functions, and identifying the correct technique to use. It is important to carefully analyze the integral and choose the best approach.

## What are some real-world applications of solving improper integrals?

Improper integrals have many real-world applications, including in physics, engineering, and economics. For example, they can be used to calculate the center of mass of an object, the work done by a variable force, or the value of an infinite series. They are also used in probability and statistics to calculate areas under curves and expected values.

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