When to use Integral Substitution? $e^x$$(1+e^x)^{1\over{2}}$
why can't use integration by part, 
What is meant by in the form of f(g(x))g'(x)?
Can you give a few example?
Thank you

$$\int (e^x+1)^{1/2}e^x\ dx=\int (1+\color{green}{e^x})^{1/2}\ d\color{green}{e^x}=\int(1+\color{green}{u})^{1/2}\ d\color{green}{u}=\frac23(1+u)^{3/2}=\frac23(1+e^x)^{3/2}.$$
Equivalenty,
$$\int (e^x+1)^{1/2}e^x\ dx=\int (\color{green}{1+e^x})^{1/2}\ d(\color{green}{1+e^x})=\int \color{green}{u}^{1/2}\ d\color{green}{u}=\frac23u^{3/2}=\frac23(1+e^x)^{3/2}.$$
Note that $dg(x)$ is the same as $g'(x)\ dx$.
