Evaluate the integral integrate of (e^(3x))/((e^x+1))dx

To evaluate the integral, we have a substitution.

`int{e^(3x)}/{e^x+1}dx`

`=int{(e^x)^3}/{e^x+1}dx`   let `u=e^x` , so `du=e^xdx`

`=int u^2/{u+1}du`   now let `v=u+1` so `dv=du`

`=int(v-1)^2/vdv`   expand numerator

`=int{v^2-2v+1}/vdv`  simplify

`=int(v^2/v-{2v}/v+1/v)dv`  simplify further

`=int(v-2+1/v)dv`  now integrate with power rule

`=1/2v^2-2v+lnv+K`  where K is the constant of integration

`=1/2(u+1)^2-2(u+1)+ln(u+1)+K`

`=1/2u^2-u-1/2+ln(u+1)+K`

`=1/2u^2-u+ln(u+1)+C`  where C=K+1 is a constant

`=1/2e^{2x}-e^x+ln(e^x+1)+C`

The integral evaluates to `1/2e^{2x}-e^x+ln(e^x+1)+C` .