Calculate $\int_{-\infty}^{\infty}ye^{-\frac{y^2}{2}+xy}dy$

Question

How to calculate $\int_{-\infty}^{\infty}ye^{-\frac{y^2}{2}+xy}dy$ ? (x and y are both real)

We know $\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}dx=1$, but how to apply this integral?

Answer 1

Hint

$$\int_{-\infty}^{\infty}ye^{-\frac{y^2}{2}+xy}dy=\int_{-\infty}^{\infty}ye^{-\frac{(y-x)^2}2}e^{\frac {x^2} 2} dy$$

Use the substition $t=y-x$ then you should be able to finish.

Spoilers :

$$\int_{-\infty}^{\infty}ye^{-\frac{y^2}{2}+xy}dy=\int_{-\infty}^{\infty}(t+x)e^{-\frac{t^2}2}e^{\frac {x^2} 2} dy=\int_{-\infty}^{\infty}xe^{-\frac{t^2}2}e^{\frac {x^2} 2} dy=xe^{\frac {x^2} 2}\int_{-\infty}^{\infty}e^{-\frac{t^2}2} dy$$