If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# 复习分步积分法

## 什么是分部积分？

$\int \phantom{\rule{-0.167em}{0ex}}\phantom{\rule{-0.167em}{0ex}}u\left(x\right){v}^{\prime }\left(x\right)dx=u\left(x\right)v\left(x\right)-\int \phantom{\rule{-0.167em}{0ex}}\phantom{\rule{-0.167em}{0ex}}{u}^{\prime }\left(x\right)v\left(x\right)dx$

## 第一套练习题：不定积分的分部积分

$\int x\mathrm{cos}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=\int u\phantom{\rule{0.167em}{0ex}}dv$
$u=x$ 表示 $du=dx$.
$dv=\mathrm{cos}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$ 代表 $v=\mathrm{sin}\left(x\right)$.

$\begin{array}{rl}\int x\mathrm{cos}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx& =\int u\phantom{\rule{0.167em}{0ex}}dv\\ \\ & =uv-\int v\phantom{\rule{0.167em}{0ex}}du\\ \\ & =x\mathrm{sin}\left(x\right)-\int \mathrm{sin}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx\\ \\ & =x\mathrm{sin}\left(x\right)+\mathrm{cos}\left(x\right)+C\end{array}$

$\int x{e}^{5x}dx=?$

## 第二套练习题：定积分的分部积分

$u=x$ 表示 $du=dx$.
$dv={e}^{-x}\phantom{\rule{0.167em}{0ex}}dx$ 代表 $v=-{e}^{-x}$.

$\begin{array}{rl}& \phantom{=}{\int }_{0}^{5}x{e}^{-x}\phantom{\rule{0.167em}{0ex}}dx\\ \\ & ={\int }_{0}^{5}u\phantom{\rule{0.167em}{0ex}}dv\\ \\ & =\left[uv{\right]}_{0}^{5}-{\int }_{0}^{5}v\phantom{\rule{0.167em}{0ex}}du\\ \\ & =\left[-x{e}^{-x}{\right]}_{0}^{5}-{\int }_{0}^{5}-{e}^{-x}\phantom{\rule{0.167em}{0ex}}dx\\ \\ & =\left[-x{e}^{-x}-{e}^{-x}{\right]}_{0}^{5}\\ \\ & =\left[-{e}^{-x}\left(x+1\right){\right]}_{0}^{5}\\ \\ & =-{e}^{-5}\left(6\right)+{e}^{0}\left(1\right)\\ \\ & =-6{e}^{-5}+1\end{array}$