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# System prepended metadata

title: Integral formula

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# 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}

