分部积分法

分部积分公式

\[\begin{align} 设：u(x),u(v)具有连续的导数，则\int udv=uv-\int vdu \end{align} \]

\[【例】计算不定积分I=\int \arctan xdx\\ \begin{align} &=\frac12\int\arctan xd(x^2+1)\\ &=\frac12(1+x^2)\arctan x-\frac12x+C \end{align} \]

\[【例】计算不定积分\int\arcsin xdx\\ \begin{align} &=x\cdot \arcsin x-\int\frac x{\sqrt{1-x^2}}dx\\ &=x\cdot \arcsin x+\int \frac1{2\sqrt{1-x^2}}d(1-x^2)\\ &=x\cdot \arcsin x+\sqrt{1-x^2}+C \end{align} \]

\[\begin{align} 【例】计算不定积分\int\sec^3xdx\\ &=\int\sec^2x\cdot\sec xdx\\ &=\int \sec xd\tan x\\ &=\sec x\cdot \tan x-\int \tan x\cdot \sec x\cdot \tan xdx\\ &=\sec x\cdot \tan x-\int\sec^3xdx+\int \sec xdx\\ \Rightarrow 2\int \sec^3xdx=\sec x\cdot \tan x+\ln|\sec x+\tan x|\\ \Rightarrow \int \sec^3xdx=\frac12\cdot\tan x+\frac12\ln|\sec x+\tan x|+C \end{align} \]