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

The main contributions to the integral will be from the interval $[0,1/\sqrt{n}]$. In this region $ \cos(x/n)\approx 1$ and therefore
$$
I_n\sim\log(3)\int_0^{1/\sqrt{n}}dx e^{-n\log(1+x^2)}(1+nx^2)\sim\log(3)\int_0^{1/\sqrt{n}}dx e^{-n x^2}(1+nx^2)
$$
because we are only introducing an exponentially small error by pushing the limits of integration up to infinity (Laplace method) we get

$$
I_n\sim\log(3)\int_0^{\infty}dx e^{-n x^2}(1+nx^2)=\log(3)\frac{3\sqrt{\pi}}{4\sqrt{n}}
$$

and the limit is $0$.
Note that  it is important to keep $1$ as well as $nx^2$ in the integrand because their contributions turn out to be of the same order!
