# Evaluate Fourier Integral

1. Dec 9, 2012

### Fusiontron

1. The problem statement, all variables and given/known data

Show that

integral from 0 - > infinity (cos(alpha*x)/(alpha^2 + 1))dalpha = (pi/2)exp(-x)

2. Relevant equations

3. The attempt at a solution

cos(alpha*x) = (1/2)(exp(i*alpha*x)+exp(-i*alpha*x))

Really don't know where to go from here.

2. Dec 9, 2012

### Dick

Use contour integrals and the residue theorem. A lot of Fourier integrals need to be done that way.

3. Dec 9, 2012

### Fusiontron

The problem says to evaluate directly with the Fourier Integral Theorem.

4. Dec 9, 2012

### Dick

Ok, so what is the Fourier Integral Theorem?

5. Dec 9, 2012