# Integral formula \begin{align} \int_0^\infty x^n e^{-x} dx &= n! \\ \int_0^{\infty} x^{2n}\ e^{-x^2} dx &= \frac{2n-1}{2}\frac{2n-3}{2}\cdots \frac{1}{2}\int_0^{\infty} e^{-x^2} dx\\ &= \frac{2n-1}{2}\frac{2n-3}{2}\cdots \frac{1}{2} \frac{\sqrt{\pi}}{2}\\ \int_0^{\infty} x^{2n+1}\ e^{-x^2} dx &= n!\int_0^{\infty} xe^{-x^2} dx\\ &= n! \ \frac{1}{2}\\ \int_0^\pi \cos^n \theta d\theta &= \frac{n-1}{n}\frac{n-3}{n-2}\cdots \int_0^\pi \cos\theta d\theta\\ \int_0^\pi \sin^n \theta d\theta &= \frac{n-1}{n}\frac{n-3}{n-2}\cdots \int_0^\pi \sin\theta d\theta\\ \int_0^{2\pi} \cos\theta\ e^{i\theta} d\theta &= \frac{\pi}{2} \\ \int_0^{2\pi} \sin\theta\ e^{i\theta} d\theta &= \frac{\pi}{2}i \\ \end{align}