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I want to compute the integral $$2\pi\int f(x) \sqrt{1+f'(x)^2} dx$$ where $f(x)=\dfrac{1}{e^x}$.

I used maple and I found that the answer is: $$\pi e^{-2x} \left[e^{2x} \arctan\left(\sqrt{e^{2x}-1}\right) - \sqrt{e^{2x}-1}\right] $$ but I can't find a way to prove it on the paper. Any help would be apreciated.

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    belial: to enclose the square root of an expression, use braces: `sqrt{expression}`. Let me know if it got the parentheses wrong in your second expression!2012-12-04
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    oh, that looks much better now! thank you!2012-12-04

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