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# 复习反向幂规则

## 什么是反向幂法则？

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

## 多项式积分

$\begin{array}{rl}\int 3{x}^{7}\phantom{\rule{0.167em}{0ex}}dx& =3\left(\frac{{x}^{7+1}}{7+1}\right)+C\\ \\ & =3\left(\frac{{x}^{8}}{8}\right)+C\\ \\ & =\frac{3}{8}{x}^{8}+C\end{array}$

$\int 14t\phantom{\rule{0.167em}{0ex}}dt=?$

## 负幂的积分

$\begin{array}{rl}\int \frac{1}{{x}^{2}}\phantom{\rule{0.167em}{0ex}}dx& =\int {x}^{-2}\phantom{\rule{0.167em}{0ex}}dx\\ \\ & =\frac{{x}^{-2+1}}{-2+1}+C\\ \\ & =\frac{{x}^{-1}}{-1}+C\\ \\ & =-\frac{1}{x}+C\end{array}$

$\int 8{t}^{-3}\phantom{\rule{0.167em}{0ex}}dt=$

## 小数幂和根号的积分

$x$被升到分数幂或根号，反向幂法则让我们可以计算它的积分。例如，考虑$\sqrt{x}$的积分：
$\begin{array}{rl}\int \sqrt{x}\phantom{\rule{0.167em}{0ex}}dx& =\int {x}^{{}^{\frac{1}{2}}}\phantom{\rule{0.167em}{0ex}}dx\\ \\ & =\frac{{x}^{{}^{\frac{1}{2}+1}}}{\frac{1}{2}+1}+C\\ \\ & =\frac{{x}^{{}^{\frac{3}{2}}}}{\frac{3}{2}}+C\\ \\ & =\frac{2\sqrt{{x}^{3}}}{3}+C\end{array}$

$\int 4{t}^{\frac{1}{3}}\phantom{\rule{0.167em}{0ex}}dt=?$