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...
$$I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.$$

Note that after enforcing the substitution $x\to x/\sqrt{n}$, we have 
$$\begin{align}
\left|\int_0^1 \frac{1+nx^2}{(1+x^2)^n}\log\left(2+\cos\left(\frac{x}{n}\right)\right)\,dx\right|&=\frac{1}{\sqrt{n}}\int_0^\sqrt{n} \frac{1+x^2}{(1+x^2/n)^n}\log\left(2+\cos\left(\frac{x}{n^{3/2}}\right)\right)\,dx\\\\
&\le \frac{\log(3)}{\sqrt{n}} \int_0^\infty \frac{1+x^2}{\left(1+\frac{x^2}{n}\right)^n}\,dx \tag 1\\\\
&\le \frac{\log(3)}{\sqrt{n}} \int_0^\infty \frac{1+x^2}{\left(1+\frac{x^2}{2}\right)^2}\,dx \tag 2\\\\
&=\frac{\log(3)}{\sqrt{n}}\,\left(\frac{3\sqrt{2}\pi}{4}\right)\tag 3
\end{align}$$
where we used the fact that $\left(1+\frac {x^2}n\right)^n\ge \left(1+\frac{x^2}{2}\right)^2$ for $n\ge2$ in going from $(1)$ to $(2)$.  
The right-hand side of $(3)$ clearly approaches zero as $n\to \infty$.  Therefore, the squeeze theorem guarantees that the integral of interest approaches zero also. And we are done!
