$\int_{-\infty}^{\infty}x e^{-x^2 + x(i+1)}dx$ how to? I am having some problems in solving the integral reported in the title, which is:
$$\int_{-\infty}^{\infty}x e^{-x^2 + x(i+1)}dx$$
As from the general theory of Gaussian integrals I have been trying to write the known integral
$$\int_{-\infty}^{\infty} e^{-x^2  - i k x}dx, \; k \in \mathbb{R}$$
which has a solution with contour integration knowing that the complex function $e^{-z^2}$ is holomorphic on the whole complex plane. Despite my efforts this route does not seem to bring to any result. 
Thanks to anyone who is so keen to try and help me.

Hint:
$$\int_{x=-\infty}^\infty xe^{-x^2+(1+i)x}dx\\
=\int_{x=-\infty}^\infty\left(x-\frac{1+i}2\right)e^{((1+i)/2)^2-(x-(1+i)/2)^2}dx
+\frac{1+i}2\int_{x=-\infty}^\infty e^{((1+i)/2)^2-(x-(1+i)/2)^2}dx\\
=e^{i/2}\int_{x=-\infty}^\infty ze^{-z^2}dz
+\frac{1+i}2e^{i/2}\int_{x=-\infty}^\infty e^{-z^2}dz.$$
The first integral vanishes by parity and the second is a more classical Gaussian integral.
