$$ \int \sin^{n} x , dx = -\frac{1}{n}\sin^{n-1}x \cos x + \frac{n-1}{n}\int \sin^{n-2}x , dx $$ $$ \int cos^{n} x , dx =\frac{1}{n}\cos^{n-1}x\sin x+\frac{n-1}{n}\int cos^{n-2}x, dx $$ $$ \int \tan^{n}x , dx = \frac{\tan^{n-1}x}{n-1} - \int \tan^{n-2}x , dx $$ $$ \int \cot^{n}x , dx = \frac{\cot^{n-1}x}{n-1} - \int \cot^{n-2}x,dx $$ $$ \int \sec^{n}x,dx=\frac{1}{n-1}\sec^{n-2}x\tan x + \frac{n-2}{n-1}\int \sec^{n-2}x,dx $$
根据前两个公式可以推出华里士公式。
(2018·中大·期末考·高数·节选) 求 $\int_{0}^{1} \sqrt{1+x^2},dx$.
法一(高阶三角函数降次)： $$\begin{align} 原式 & =\int_{0}^{1} \sec x , d\tan x \ & =\int_{0}^{\frac{\pi}{4}} \sec ^{3}x , dx \ & =\frac{1}{2}\sec x\tan x\left|{0}^{\frac{\pi}{4}}\right.+\int{0}^{\frac{\pi}{4}} \sec x , dx \ & =\frac{\sqrt{ 2 }}{2}+\ln \left| \sec x+\tan x \right| \left|_{0}^{\frac{\pi}{4}} \right. \ & =\frac{\sqrt{ 2 }}{2}+\ln(\sqrt{ 2 }+1) \end{align}$$ 法二(配凑＋原积分重现)：