Let $f: \mathbb R\rightarrow \mathbb R$ satisfying \begin{equation*} f(x) + f\bigg(1-{1\over x}\bigg) = \arctan x\end{equation*} for all $x\neq 0$. I want to evaluate \begin{equation*} \int_0^1 f(x) dx \end{equation*}
I know that $\tan\bigg(f(x) + f\bigg(1-{\cfrac 1x}\bigg)\bigg) = x$ and $-\pi/2 < f(x) + f\bigg(1-{1\over x}\bigg) < \pi/2$, but none of this seems useful yet. Since $x\neq 0$, the answer should be the solution to \begin{equation*} \lim_{n\rightarrow 0}\int_n^1f(x)dx\end{equation*} But again, I'm still unsure where to go from here. Any hint would be appreciated.