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 (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 (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}`