Show that $\lim_{n\to\infty}\int_{[0,1]}\frac{nx}{1+n^2x^2}=0$.

Question

$\lim_{n\to\infty}\int_{[0,1]}\frac{nx}{1+n^2x^2}=0$ is to be shown. This should be done using the Lebesgue Dominated Convergence theorem. I can see that the sequence of functions $(f_n)$, where $f_n(x)=\frac{nx}{1+n^2x^2}$ converges pointwise to zero on $[0,1]$. I need help in figuring out a dominating function to apply the aforementioned theorem, i.e. a function $g$ such that it is integrable on $[0,1]$ and $|f_n|\leq g$ a.e. on $[0,1]$ for all $n$. Someone please give me a hint. Thanks.