Find the limits $\lim\_{n \to \infty} \int\_0^1 \frac{\ln(1+x)}{1+nx^2}$

This problem was given at the end of Lebesgue Integration Chapter. I assume I have to use use Lebesgue Convergence Theorem. In other words I have to find a function $g$ that dominates $f \_n = \frac{\ln(1+x)}{1+nx^2)}$ then prove that $f\_n$ converge to a function $f$. If I prove this then $\lim\_{n \to \infty} \int\_0^1 \frac{\ln(1+x)}{1+nx^2} = \int\_0^1 f$

Any help on finding $g$ and $f$? Thanks