Integrate $$\int_{-\infty}^{\infty} \cos(\sqrt{2} x)e^{-x^2} dx$$
All i want to know is how to proceed. I have been stuck at this for a very long time now.
I tried writing the $\cos$ term as exponentials using Euler's formula and then to integrate the product term but I don't know how to integrate that either.
All you need is the real part of $$ \int_{-\infty}^{+\infty}\exp\left[-\left(x^2-i\sqrt{2}x\right)\right]\,dx \stackrel{x\mapsto x+\frac{i}{\sqrt{2}}}{=}\int_{-\infty}^{+\infty}\exp\left[-\left(x^2+\frac{1}{2}\right)\right]\,dx=\color{blue}{\sqrt{\frac{\pi}{e}}}.$$