How to integrate this function How to integrate the following double integral:
$$\int_0^{+ \infty}\int_{- \infty}^{+ \infty}e^{(ax+by)} e^{-x} e^{-0.5(y-x)^2} \; dy \; dx$$

$$\begin{align*}
\int_0^\infty\int_{-\infty}^\infty e^{ax+by}e^{-x}e^{-0.5(y-x)^2}\ dy\ dx &= \int_0^\infty\int_{-\infty}^\infty e^{(a-1)x}e^{by}e^{-\frac12(y^2-2xy-x^2)}\ dy\ dx \\
 &= \int_0^\infty e^{(a-1)x+\frac12x^2}\int_{-\infty}^\infty e^{-\frac12y^2+(b+x)y}\ dy\ dx
\end{align*}
$$
Now, evaluate the inner integral first, treating $x$ as a constant.
