Let's solve a beautiful and tricky integral problem.
The Problem:
Let $$ I$$=$$\int e^x/(e^{4x}+e^{2x}+1) dx$$ $$ J$$=$$ \int e^{-x}/(e^{-4x}+e^{-2x}+1)dx$$. Find the value of (J-I).
Solution:
$$ I$$=$$\int e^x/(e^{4x}+e^{2x}+1) dx$$
$$J$$= $$\int e^{-x}/(e^{-4x}+e^{-2x}+1)dx$$
Let (e^x)=(z)
$$ J-I$$=$$\int\frac{e^x(e^{2x-1})}{e^{4x}+e^{2x}+1}dx$$=$$\int\frac{z^2-1}{z^4+z^2+1}dz
$$
$$ =\frac{1}{2}ln\frac{(e^x+e^-x-1)}{(e^x+e^-x+1)}+c$$
(where c is a constant of integration)
