Question: $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

Answer: 
Integration by substitution says that
$$
\int f(g(x))g'(x) \; dx = \int f(u)\;du
$$
where $u = g(x)$. This is a tool that will let you compute the integral of all functions that look exactly like $f(g(x))g'(x)$. When you first see this, you might be thinking that this seems like a very specialised rule that only seem good for very special cases. But it turns out that this is very useful.

In your example, you have
$$
\int (1 + e^x)^{1/2}e^x\; dx.
$$
If you want to know whether or not you can use substitution you just have to determine if it is possible to write $(1 + e^x)^{1/2}e^x$ as $f(g(x))g'(x)$ for some functions $f$ and $g$. Mabe by trail and error you find that indeed if $f(x) = x^{1/2}$ and $g(x) = 1+e^x$, then $f(g(x))g'(x) = (1 + e^x)^{1/2}e^x$. And so with $u = g(x) = 1+e^x$, you get
$$
\int (1 + e^x)^{1/2}e^x\; dx = \int u^{1/2} \; du = \frac{2}{3}u^{3/2} + C = \frac{2}{3}(1+e^x)^{3/2} + C.
$$

---

You ask why you you can't use integration by parts. Integration by parts says that
$$
\int f(x)g'(x)\; dx = f(x)g(x) - \int f'(x)g(x)\; dx.
$$
So you would have to find $f$ and $g$ such that $f(x)g'(x) = e^x(1+e^e)^{1/2}$. You can of course choose $f(x) = (1 + e^x)^{1/2}$ and $g(x) = e^x$, so then you get
$$
(1 + e^x)^{1/2}e^x - \int \frac{1}{2}(1+e^x)^{1/2}e^xe^x\;dx.
$$
This is perfectly fine, but it doesn't do you any good. You wanted to find one complicated integral and now you just have another complicated integral. So you *can* use integration by parts, but, again, it doesn't help you much. It would be a good exercise to try other choices of $f$ and $g$ to convince yourself that this indeed doesn't help much.
in your example i would set $t=\sqrt{1+e^{x}}$ then we get $t^2=1+e^{x}$ and we have $2tdt=e^xdx$ thus your integral is $2\int t^2dt$