I am trying to integrate $\int_{-\infty}^{\infty} x^ne^{x-x^2} dx$ for n = 0,1,2 but I can't think of how to do this for any n.
I managed (I believe) to show that $\int_{-\infty}^{\infty} xe^{x-x^2} dx = \frac{1}{2}\int_{-\infty}^{\infty} e^{x-x^2}dx$ and similarly that $\int_{-\infty}^{\infty} x^2e^{x-x^2} dx = \frac{3}{4}\int_{-\infty}^{\infty} e^{x-x^2}dx$.
This means I only need to do the integral for one n, but I can't see which or how.
If anyone could put me on the right track it would be much appreciated.
HINT:
Let $F(a)=\int_{-\infty}^\infty e^{ax-x^2}\,dx=\sqrt\pi e^{a^2/4}$. Then, we have
$$\left.\left(\frac{d^nF(a)}{da^n}\right)\right|_{a=1}=\int_{-\infty}^\infty x^n e^{x-x^2}\,dx$$