TITLE: Calculating a limit of integral
QUESTION [10 upvotes]: Computing the limit: $$\lim_{n\rightarrow\infty}\left(\frac{1}{3\pi}\int_\pi^{2\pi}\frac{x}{\arctan(nx)} \ dx\right)^n$$
I made the substiution $t=nx$ then, we have: $$I=\frac{1}{n^2}\int_{n\pi}^{2n\pi}\frac{t}{\arctan t}dt$$ where $I$ is the inside integral. How would you continue?

REPLY [5 votes]: Asymptotics of the integral:
When $x$ is large,
\begin{align}
\frac{1}{\arctan{x}}
=&\frac{2}{\pi}\frac{1}{1-\frac{2}{\pi}\arctan\left(\frac{1}{x}\right)}\tag1\\
=&\frac{2}{\pi}\frac{1}{1-\frac{2}{\pi}\left(\frac{1}{x}-\frac{1}{3x^3}+\cdots\right)}\tag2\\
=&\frac{2}{\pi}\left(1+\frac{2}{\pi}\left(\frac{1}{x}-\frac{1}{3x^3}+\cdots\right)+\frac{4}{\pi^2}\left(\frac{1}{x}-\frac{1}{3x^3}+\cdots\right)^2+\cdots\right)\tag3\\
=&\frac{2}{\pi}+\frac{4}{\pi^2 x}+\frac{8}{\pi^3 x^2}+\mathcal{O}\left(x^{-3}\right)\tag4
\end{align}
Then
\begin{align}
\frac{1}{3\pi}\int^{2\pi}_\pi\frac{x}{\arctan(nx)}{\rm d}x
=&\frac{1}{3\pi}\int^{2\pi}_\pi\left(\frac{2x}{\pi}+\frac{4}{\pi^2 n}+\frac{8}{\pi^3 n^2x}+\cdots\right)\ {\rm d}x\\
=&\ 1+\frac{4}{3\pi^2n}+\frac{8\ln{2}}{3\pi^4n^2}+\mathcal{O}\left(n^{-3}\right)
\end{align}

Computing the limit:
Your limit is thus
\begin{align}
\lim_{n\to\infty}\left(\frac{1}{3\pi}\int^{2\pi}_\pi\frac{x}{\arctan(nx)}{\rm d}x\right)^n
=&\ \exp\left\{\lim_{n\to\infty}n\ln\left(\frac{1}{3\pi}\int^{2\pi}_\pi\frac{x}{\arctan(nx)}{\rm d}x\right)\right\}\tag5\\
=&\ \exp\left\{\lim_{n\to\infty}n\left(\frac{4}{3\pi^2n}+\mathcal{O}\left(n^{-2}\right)\right)\right\}\tag6\\
=&\ \large{\color{red}{\exp\left(\frac{4}{3\pi^2}\right)}}\normalsize\approx1.144645419236050\tag7\\
\end{align}

Numerical Verification:
Using Mathematica I am getting
$$\left(\frac{1}{3\pi}\int^{2\pi}_\pi\frac{x}{\arctan(999999999x)}{\rm d}x\right)^{999999999}\approx1.144645419247325$$
which is consistent with the derived result.

Explanation:
$(1)$: Used the fact that $\displaystyle\arctan{x}=\frac{\pi}{2}-\arctan\left(\frac{1}{x}\right)$. 
$(2)$: Used the series for $\arctan{x}$. 
$(3)$: Used a geometric series. 
$(4)$: Expanded the terms. 
$(5)$: Used the fact that $\displaystyle\lim_{n\to\infty}a_n^{b_n}=\exp\lim_{n\to\infty}b_n\ln(a_n)$.
$(6)$: Used the series for $\ln(1+x)$. 
$(7)$: As $n\to\infty$, only the constant term remains.