e^2x (1+e^2x)^3 dx

in front to the "e" is the "s" like sign

\int e^{2x}(1+e^{2x})^{3}dx

Let u=e^{2x}, \;\ du=2e^{2x}dx, \;\ \frac{du}{2}=e^{2x}dx

Make the subs and get:

\frac{1}{2}\int (1+u)^{3} du

Now, it is like the other, only easier.