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Let $f$ be a monotonic differentiable function; $f: (a,b)\to \mathbb{R}$, so that F'=f.

I need to prove that:

$\int f^{-1}(x)dx=xf^{-1}(x)-F(f^{-1}(x))+C$.

I tried to use the following formula: \int u'v=uv-\int v'u+C, but I can't see how I move on from this.

Any Hints?

Thanks A Lot!

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    @N.S.: Thanks for the correction!2011-11-22

2 Answers 2

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Let $f^{-1}(x) = y$. Then $x = f(y)$ which gives us dx = f'(y) dy.

Hence, \begin{align} \int f^{-1}(x) dx & = \int y d(f(y)) & (\because y = f^{-1}(x))\\ & = y f(y) - \int f(y) dy + C & (\text{Integration by parts})\\ & = f^{-1}(x)x - F(y) + C & (\because F' = f \implies \int fdy = F)\\ & = xf^{-1}(x) - F(f^{-1}(x)) + C & (\because y = f^{-1}(x)) \end{align}

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    @Jozef: You need to know $f$ is monotonic so that $f^{-1}$ is well-defined.2011-11-22
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