1. Solving an Integral Help

$\displaystyle \int\frac{dx}{\sqrt{e^{2x}-1}}$

I'm not sure how to start this problem so any help would be appreciated.

2. Originally Posted by summermagic
$\displaystyle \int\frac{dx}{\sqrt{e^{2x}-1}}$
I didn't see anything useful, so I plugged your integral into The Integrator....

Working backwards, you can easily confirm that $\displaystyle arctan\left(\sqrt{e^{2x}\, -\, 1}\right)$ does indeed differentiate back to your integrand, but I'm not seeing how you're supposed to come up with this "going frontways"....

3. Hello, summermagic!

$\displaystyle \int\frac{dx}{\sqrt{e^{2x}-1}}$
It can be done with a Trig Substitution: .$\displaystyle e^x \,=\,\sec\theta$
. . but here is a back-door approach.

Divide top and bottom by $\displaystyle e^x$

.$\displaystyle \int\frac{\dfrac{dx}{e^x}}{\dfrac{\sqrt{e^{2x}-1}}{e^x}} \;=\;\int \frac{e^{\text{-}x}\,dx}{\sqrt{\dfrac{e^{2x}-1}{e^{2x}}}} \;=\;\int\frac{e^{\text{-}x}dx}{\sqrt{1 - e^{\text{-}2x}}}$

Let: .$\displaystyle u \:=\:e^{\text{-}x}\quad\Rightarrow\quad du \:=\:-e^{\text{-}x}\,dx \quad\Rightarrow\quad e^{\text{-}x}\,dx \:=\:-du$

Substitute: .$\displaystyle \int\frac{-du}{\sqrt{1-u^2}} \;=\;\arccos u + C$

Back-substitute: .$\displaystyle \arccos\left(e^{\text{-}x}\right) + C$