Compute $\lim_{n \to \infty}n\int_{0}^{1}\frac{\cos x}{1+e^{nx}},\ n\in\mathbb{N}$ Let 
$$
I_n = n\int_{0}^{1}\frac{\cos x}{1+e^{nx}}\,dx\,,\quad n\epsilon \mathbb N^*.
$$ Calculate $\lim_{n \to \infty}I_n$

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\begin{align}
&\bbox[10px,#ffd]{\lim_{n \to \infty}\bracks{n\int_{0}^{1}{\cos\pars{x} \over 1 + \expo{nx}}\,\dd x}} =
\lim_{n \to \infty}\bracks{n\int_{0}^{1}{\dd x \over 1 + \expo{nx}} -
n\int_{0}^{1}{1 - \cos\pars{x} \over 1 + \expo{nx}}\,\dd x}
\\[5mm] = &\
\underbrace{\int_{0}^{\infty}{\dd x \over 1 + \expo{x}}}_{\ds{\ln\pars{2}}}\ -\
\lim_{n \to \infty}\bracks{n\int_{0}^{1}{2\sin^{2}\pars{x/2} \over 1 + \expo{nx}}\,\dd x}
\\[5mm] = &\
\ln\pars{2} -
2\lim_{n \to \infty}\int_{0}^{n}{\sin^{2}\pars{x/\bracks{2n}} \over 1 + \expo{x}}\,\dd x
\end{align}

Note that
  $\ds{0 < 2\int_{0}^{n}{\sin^{2}\pars{x/\bracks{2n}} \over 1 + \expo{x}}\,\dd x < {1 \over 2n^{2}}\int_{0}^{\infty}{x^{2} \over 1 + \expo{x}}\,\dd x\,\,\,\stackrel{\mrm{as}\ n\ \to\ \infty}{\to}\,\,\, {\LARGE 0}}$

Then,
$$
\bbox[10px,#ffd]{\lim_{n \to \infty}\bracks{n\int_{0}^{1}{\cos\pars{x} \over 1 + \expo{nx}}\,\dd x}} = \bbx{\ln\pars{2}} \approx 0.6931
$$
