What is the value of $I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.$? Find the integral $I$.....it looks like a good problem which I was not able to solve ....please help...
$$I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.$$

By Bernoullis inequality, we have $\frac{1+nx^2}{(1+x^2)^n}\leq 1$.
Also, $\cos{t}\leq 1$. 
Therefore, $$|f_n(x)|=\left|{{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\right|\leq\log{3}=g(x)$$
Now, by DCT, we have
$I=0$.
