dy/sqrt(e^(2y)-1) using u-substitution

[Muteki]

dy/sqrt(e^(2y)-1)

I know i need to use a u sub, but I cannot seem to find the correct sub for this problem.

 

[soroban]

Re: dy/sqrt(e^(2y)-1)

Hello, Muteki!

\(\displaystyle \L\int\frac{dy}{\sqrt{e^{2y}\,-\,1}}\)

Let \(\displaystyle e^y \:=\:\sec\theta\;\;\Rightarrow\;\;y \:=\:\ln(\sec\theta)\;\;\Rightarrow\;\;dy \:=\:\frac{\sec\theta\tan\theta}{\sec\theta}\,d\theta\:=\:\tan\theta\,d\theta\)

. . and: \(\displaystyle \:\sqrt{e^{2y}\,-\,1}\:=\:\sqrt{\sec^2\theta\,-\,1} \:=\:\sqrt{\tan^2\theta} \:=\:\tan\theta\)

Substitute: \(\displaystyle \L\:\int\frac{\tan\theta\,d\theta}{\tan\theta} \;=\;\int d\theta \;=\;\theta\,+\,C\)

Back-substitute: \(\displaystyle \L\:\text{arcsec}\left(e^y\right)\,+\,C\)