How do you find the antiderivative of $e^(2x)/sqrt(1-e^x)$ ?

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Answer 1 · 2016-09-09T21:55:36

$(-2(e^x+2)sqrt(1-e^x))/3+C$

We have:

$I=inte^(2x)/sqrt(1-e^x)dx$

Let $u=e^x$ . This implies that $du=e^xdx$ . We can write $e^(2x)$ as $e^x(e^x)$ :

$I=int(e^x(e^x)dx)/sqrt(1-e^x)=intu/sqrt(1-u)du$

Letting $v=1-u$ , such that $dv=-du$ , and manipulating to show that $u=1-v$ :

$I=-int(1-v)/sqrtvdv=int(v^(1/2)-v^(-1/2))dv$

Integrating using the $intv^ndv=v^(n+1)/(n+1),n!=-1$ rule:

$I=v^(3/2)/(3/2)-v^(1/2)/(1/2)=2/3v^(3/2)-2v^(1/2)=(2sqrtv(v-3))/3$

Since $v=1-u$ , and $u=e^x$ , so $v=1-e^x$ :

$I=(2sqrt(1-e^x)(1-e^x-3))/3=(-2(e^x+2)sqrt(1-e^x))/3+C$

How do you find the antiderivative of e^(2x)/sqrt(1-e^x)e^(2x)/sqrt(1-e^x)?