What is the solution to #int e^x/(1 - e^(2x))^(1/2) dx#?

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Answer 1 · 2017-03-03T17:59:08

#int e^x/(1 - e^(2x))^(1/2)dx = arcsin(e^x) + C#

We can write this as

#int e^x/sqrt(1 - (e^x)^2)#

It becomes clear that a u-subsittution of #u = e^x# would be beneficial. This means that #du = e^xdx# and #dx= (du)/e^x#.

#int e^x/sqrt(1 - u^2) * (du)/e^x#

#int 1/sqrt(1 - u^2) du#

This is a known integral.

#arcsinu +C#

#arcsin(e^x) +C#

Hopefully this helps!