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# 复习常规积分

## 多项式

$\int {x}^{n}\phantom{\rule{0.167em}{0ex}}dx=\frac{{x}^{n+1}}{n+1}+C$

## 根基

$\begin{array}{rl}\int \sqrt[m]{\phantom{A}{x}^{n}}\phantom{\rule{0.167em}{0ex}}dx& =\int {x}^{{}^{\frac{n}{m}}}\phantom{\rule{0.167em}{0ex}}dx\\ \\ & =\frac{{x}^{{}^{\frac{n}{m}+1}}}{\frac{n}{m}+1}+C\end{array}$

## 三角函数

$\int \mathrm{sin}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=-\mathrm{cos}\left(x\right)+C$
$\int \mathrm{cos}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=\mathrm{sin}\left(x\right)+C$
$\int {\mathrm{sec}}^{2}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=\mathrm{tan}\left(x\right)+C$
$\int {\mathrm{csc}}^{2}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=-\mathrm{cot}\left(x\right)+C$
$\int \mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=\mathrm{sec}\left(x\right)+C$
$\int \mathrm{csc}\left(x\right)\mathrm{cot}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=-\mathrm{csc}\left(x\right)+C$

## 指数函数

$\int {e}^{x}\phantom{\rule{0.167em}{0ex}}dx={e}^{x}+C$
$\int {a}^{x}\phantom{\rule{0.167em}{0ex}}dx=\frac{{a}^{x}}{\mathrm{ln}\left(a\right)}+C$

## 对数函数的积分

$\int \frac{1}{x}\phantom{\rule{0.167em}{0ex}}dx=\mathrm{ln}|x|+C$

## 反三角函数的积分

$\int \frac{1}{\sqrt{{a}^{2}-{x}^{2}}}\phantom{\rule{0.167em}{0ex}}dx=\mathrm{arcsin}\left(\frac{x}{a}\right)+C$
$\int \frac{1}{{a}^{2}+{x}^{2}}\phantom{\rule{0.167em}{0ex}}dx=\frac{1}{a}\mathrm{arctan}\left(\frac{x}{a}\right)+C$