Highly Useful Formulas

hfwang132@gmail.com (hfwang132) — Sat, 01 Jan 2022 19:21:09 +0800

The following formulas can make problem solving much more convenient.

\[I_n = \int_{0}^{\infty}x^n e^{-b x^2}\mathrm{d}x %= %\left\{\begin{aligned} %\end{aligned}\right. \]

Here is a convenient way to remember it:

First, remember

\[I_0 = \frac{1}{2}\sqrt{\frac{\pi}{b}},\quad I_1 = \frac{1}{2}\frac{1}{b}\]

Next,

\[I_{n+2}=\frac{n+1}{2b}I_n\]

This formula applies to both odd and even cases of \(n\).

If you are dealing with Gaussian functions (for example, in probability theory), then this integral will appear very frequently!

\[I_n=\int_{0}^{\frac{\pi}{2}}\sin^{n}x\mathrm{d}x = \int_{0}^{\frac{\pi}{2}}\cos^{n}x\mathrm{d}x\]

The following expression gives \(I_n\)

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