Evaluate this (complicated) integral in two variables

[kingwinner]

1. The problem statement, all variables and given/known data
I was doing a statistics problem, and ended up with this integral:
∞
∫ [1/(2pi)] [e-(w2+z2w2)/2] |w| dw
w=-∞
How can I evaluate this?

2. Relevant equations
3. The attempt at a solution
1) How can we integrate |w| (with the absolute value)?
2) What should we do with the z? Can we treat it as a constant here?
3) Do we need integration by parts here? If so, what should u and dv be? Looks like the thing has no antiderivative...

Thanks for helping!

[HallsofIvy]

1) Do what you should have learned to do when you first learned about absolute value: break it into "w< 0" and "w> 0". If w< 0, |w|= -w. If w> 0, |w|= w. Do two separate integrals and then add.

2) Yes, since you are not integrating with respect to z, you are looking for a value for each z and should treat it as a constant as far as the integral with respect to w is concerned.

3) Looks to me like a simple substitution. Let y= (1+ z)w2.