Continuity and integration using Dominated Convergence theorem I have to use the Dominated Convergence Theorem to show that $\lim \limits_{n \to \infty}$ $\int_0^1f_n(x)dx=0$ where $f_n(x)=\frac{n\sqrt{x}}{1+n^2x^2}$.  
I did the following:
$$\frac{n\sqrt{x}}{1+n^2x^2} <\frac{n\sqrt{x}}{n^2x^2} = \frac{x^{-\frac{3}{2}}}{n}\leq x^{-\frac{3}{2}} $$
But $$\int_0^1x^{-\frac{3}{2}}dx=\frac{-2}{\sqrt{x}}\biggr|_0^1$$ which doesn't seem right. Any help will be appreciated. 

Use for example:
$$\frac{n\sqrt x}{1+n^2x^2}=\frac{1}{\sqrt x}\cdot\frac{1}{\frac{1}{nx}+nx}\le \frac{1}{2\sqrt x}$$
since $u+\frac{1}{u}\ge 2$ for $u>0$
And $\int_0^1\frac{1}{2\sqrt x}=1<\infty$
(this is equivalent to AM-GM inequality on the denominator $1+n^2x^2\ge2nx$)
