# Double integral using the dirac delta

1. May 21, 2009

### plaes

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

Need to integrate using the dirac delta substitution:
$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\!x^2\cos(xy)\sqrt{1-k^2\sin^2(y)}\, dx\, dy$$

2. Relevant equations
$$\cos(xy) = \frac{1}{2}\left(e^{ixy} + e^{-ixy}\right)$$

$$\delta\left[g(t)\right] = \frac{1}{2\pi}\int_{-\infty}^{\infty}\!e^{ikg(t)}\,dk$$

3. The attempt at a solution

1) First I tried replacing cos with the exponents, this allowed breaking the integral into two (almost identical ;) ) parts.
2) Next I should use the second formula (the one with delta) and replace exp with delta, which would help me to get rid of the x-parts...

but the problem is how can I substitute delta when I have something like this (how to deal with the x^2 ???):
$$\frac{1}{2\pi}\int_{-\infty}^{\infty}\!x^2e^{ix(y)}\,dx$$