There maybe a simpler and more direct method, but you may need to make use of the following formula:

Then you may obtain the following:

where are the CDF of respectively.

Next you may show that the integrand tends to zero when it goes to infinity (property of CDF) and has a unique global and local maximum at by usual calculus check with the given property. Thus showing that the integrand, and therefore the integral is non-negative which completes the proof.