$f^{-1}$的原函数

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$F$为$f$的原函数,则\[\int f^{-1}(x)\, dx = x \cdot f^{-1}(x)-(F \circ f^{-1})(x)+C.\]
例\begin{aligned}
\displaystyle \int \arcsin(x)\, dx
&= x \cdot \arcsin(x)- (-\cos\circ \arcsin)(x)+C \\
&=x \cdot \arcsin(x)+\sqrt{1-x^2}+C.
\end{aligned}\begin{aligned}
\int \log(x)\, dx
&= x \cdot \log(x)-(\exp \circ \log)(x) + C \\
&= x \cdot \left( \log(x)-1 \right) + C.
\end{aligned}