Integration multi dimension exp function

Question

I faced this problem, i tried to write it in 1 dimension. Thanks for any help. $$ \int_{{R}^d}e^{t^T\alpha}t^T\alpha e^{-\gamma ||t||^2}dt~~where~~ t,\alpha \in \mathbb{R}^d ~~~\gamma \in \mathbb{R} $$

Answer 1

Let introduce integral $$ I(\lambda)= \int_{{R}^d}e^{\lambda t^T\alpha} e^{-\gamma ||t||^2}dt\,. $$ You integral is a derivetive of this one. In fact it is a Gaussian integral. You can modify it in the following way $$ I(\lambda)=\int_{{R}^d}\exp\left[-\gamma ||t-\frac{\lambda}{2\gamma}\alpha||^2+\frac{\lambda^2}{4\gamma}||\alpha||^2\right]dt\,\\=\frac{\pi^{D/2}}{\gamma^{D/2}} \exp\left[\frac{\lambda^2}{4\gamma}||\alpha||^2\right] $$ You result is a $I'(1)=\dfrac{1}{2\gamma}\dfrac{\pi^{D/2}}{\gamma^{D/2}} \exp\left[\dfrac{1}{4\gamma}||\alpha||^2\right]$