Integral Involving Gaussian Q function and exponential
Solution 1:
The paper by Y. Chen and N.C. Beaulieu, "Solutions to Infinite Integrals of Gaussian Q-Function Products and Some Applications," in IEEE Communications Letters discusses the integrals of the form
$\displaystyle \int_{0}^{\infty}x^{a}e^{-bx^{2}}Q(cx) dx$.
May be it can help.