Question

E^(x/2) /(e^(x/3) +1)

Answer 1

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

start with substituting z=e^x

$z=e^x\\dz=e^x dx\\\int \frac{dz}{z^{5/6}+z^{1/2}}$

then substitute u = z^(1/6):

$u=z^{1/6}\\dz=6 z^{5/6}du\\6 \int \frac{u^2}{u^2+1}du$

that integral has a standard form that can be looked up in integral tables, it has the following solution:

$6(u - \tan^{-1} u) + \mbox{constant}$

substituting back the the variable z and then x you get the final solution:

$6 e^{x/6} - 6 \tan^{-1}(e^{x/6}) + \mbox{constant}$