How do you find the antiderivative of #e^(2x)/sqrt(1-e^x)#?
1 Answer
Sep 9, 2016
Explanation:
We have:
#I=inte^(2x)/sqrt(1-e^x)dx#
Let
#I=int(e^x(e^x)dx)/sqrt(1-e^x)=intu/sqrt(1-u)du#
Letting
#I=-int(1-v)/sqrtvdv=int(v^(1/2)-v^(-1/2))dv#
Integrating using the
#I=v^(3/2)/(3/2)-v^(1/2)/(1/2)=2/3v^(3/2)-2v^(1/2)=(2sqrtv(v-3))/3#
Since
#I=(2sqrt(1-e^x)(1-e^x-3))/3=(-2(e^x+2)sqrt(1-e^x))/3+C#