How to find $\int{\frac{x}{\sqrt{x^2+1}}dx}$? I started simplifying  $$\int{\dfrac{x}{\sqrt{x^2+1}}dx}$$
but I always get this: 
$$\int{x(x^2+1)^{-1/2}dx}.$$
But I don't know how to follow by that way.

Personally, I dislike the use of variable substitution, which is sort of mechanical, for problems that can be solved by applying concept. Not to mention that changing variables is always taken extremely lightly, as if we can just plug in any expression for $x$ and presto! new variable! For example $u = x^2+1$ is clearly not valid in all of $\mathbb R$ as it isn't injective, but it is in $\mathbb R^+$, which works out because that's all we need in this case, but no one seemed to care to use this justification! Also in for beginner calculus students, "$\mathrm dx$" means god knows what, all I need to know is that $\mathrm d(f(x))=f'(x)\mathrm dx,$ and hence we encourage liberal manipulation of meaningless symbols.
For these kinds of integrals just use the chain rule: $$
(f(u(x))' = f'(u(x))u'(x)\Rightarrow f(u(x))=\int f'(u(x))u'(x) \mathrm dx$$
So here just identify $u(x)=x^2+1$ and $u'(x)=2x$, so all we need is a factor of $2$ inside the integral which we can obtain if we also divide by $2$, which we can then factor out. I think this is a very good method in general, that is, the method of looking for when a function and its own derivative are present in an integral.
