# How to work out the integral $\int e^x·\sqrt{1+e^x}$

$\int e^x·\sqrt{1+e^x}\,dx$

Can someone explain to me how to work this integral out?
I tried integration by parts, however, that does not give a clear answer.
I got a hint from our professor that we should use substitution, but if I substitute "$1+e^x$" for example, I still can't calculate it by integration by parts...

• let $1+e^x=u$. . Nov 26, 2017 at 14:42

Substitute $1+e^{ x }=u$ then $du={ e }^{ x }dx$

$$\\ \\ \int e^{ x }\sqrt { 1+e^{ x } } \,dx=\int \sqrt { u } \,du=\frac { 2u\sqrt { u } }{ 3 } +C=\\ =\frac { 2 }{ 3 } { \left( { e }^{ x }+1 \right) }^{ \frac { 3 }{ 2 } }+C$$

• It's $\frac23(1+e^x)^\frac32+C$ not $\frac32(e^x)^\frac32+C$. Nov 26, 2017 at 14:47
• @TheSimpliFire,it was a typo,thank you Nov 26, 2017 at 14:53
• Note also the inversion of $\frac32$ :) Nov 26, 2017 at 14:55
• today is not my day ( Nov 26, 2017 at 15:00
• You forgot a : in your smily LOL
– user223391
Nov 26, 2017 at 15:03

Do the substitution $1+e^x=t$ and $e^x\,\mathrm dx=\mathrm dt$.