Evaluating a "Fourier Transform" Integral 1. The problem statement, all variables and given/known data Evaluate I = ∫[0,∞] e-ktw2 cos(wx) dw in the following way: Determine ∂I/∂x, then integrate by parts. 2. Relevant equations Possibly? 3. The attempt at a solution Since integral limits do not depend on x, the partial with respect to x should simply be: I = ∫[0,∞] e-ktw2 cos(wx) (-w) dw The integration by parts poses the main problem. I have done a change of variables allowing z = w2, although it seems I will have a recursion issue with an extra integral that is unable to be evaluated after each integration by parts. For integration by parts, I previously let u = sin(x√z) and dv = e-ktzdz but this doesn't seem to lead anywhere good. I considered using Euler's formula to replace the cosine but this seems to lead in the wrong direction also. Any suggestions are appreciated. Thanks!