Difficult Integral Question

Question

I'm trying to evaluate the following integral;

$$\int e^{(x^2 - z^2)} (2x \cos(2xz) - 2z \sin(2xz)) dz$$

I've tried splitting it up, and using integration by parts, but it just isn't coming out in a simple way. I've been stuck on this for hours. I'm sure there's some rule or trick I can use, but I'm really not sure.

Any assistance would be fantastic. :)

Answer 1

Hint: Observe that the first $e^{x^2}$ factors out of the integral. After removing it, notice that what you have looks sort of like the output of a product rule. Can you work backwards to find the product?

Answer 2

Hint: Distribute the product and split into two integrals. Do nothing with the first, and integrate the second by parts, using $u = \sin(2xz)$.