How can the Fourier Integral Theorem be used to evaluate improper integrals?

[Fusiontron]

Homework Statement

Show that

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

Homework Equations

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.

 

[Dick]

Homework Statement

Show that

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

Homework Equations

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.

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

 

[Fusiontron]

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

 

[Dick]

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

Ok, so what is the Fourier Integral Theorem?

 

[Fusiontron]

Well this looks promising:

http://math.stackexchange.com/quest...al-theorem-to-evaluate-the-improper-integrals