[tex]\int e^{x/2} /(e^{x/3} +1) dx[/tex]
start with substituting z=e^x
[tex]z=e^x\\dz=e^x dx\\\int \frac{dz}{z^{5/6}+z^{1/2}}[/tex]
then substitute u = z^(1/6):
[tex]u=z^{1/6}\\dz=6 z^{5/6}du\\6 \int \frac{u^2}{u^2+1}du[/tex]
that integral has a standard form that can be looked up in integral tables, it has the following solution:
[tex]6(u - \tan^{-1} u) + \mbox{constant}[/tex]
substituting back the the variable z and then x you get the final solution:
[tex]6 e^{x/6} - 6 \tan^{-1}(e^{x/6}) + \mbox{constant}[/tex]
