int e^(3x)/(e^x+e^(3x)) dx

int e^(3x)/(e^x+e^(3x))dx
To solve this, let's simplify first the integrand.
=int e^(3x)/(e^x(1+e^(2x)))dx
= int (e^x * e^(2x))/(e^x(1+e^(2x)))dx
= int e^(2x)/(1+e^(2x))dx
Then, apply u-substitution method. 
u=1+e^(2x)
du = e^(2x)*2dx
(du)/2=e^(2x)dx
Expressing the integral in terms of u, it becomes:
= int 1/(1+e^(2x)) * e^(2x)dx
= int 1/u * (du)/2
= 1/2 int 1/u du
=1/2ln|u|+ C
And, substitute back u = 1+e^(2x) .
=1/2ln|1+e^(2x)|+C
 
Therefore, int e^(3x)/(e^x+e^(3x))dx = 1/2ln|1+e^(2x)| + C .