Wiki-Math-FL

\begin{align*} \frac{d}{du} \int\limits_{-u}^u \arctan\left(\frac{1}{x}\right) dx &= \frac{d}{du} \left(\,\int\limits_{-u}^0 \arctan\left(\frac{1}{x}\right) dx + \int\limits_{0}^u \arctan\left(\frac{1}{x}\right) dx \right) \\ &= \frac{d}{du} \left(-\int\limits_{0}^{-u} \arctan\left(\frac{1}{x}\right) dx + \int\limits_{0}^u \arctan\left(\frac{1}{x}\right) dx \right) \\ &= -\arctan\left(-\frac{1}{u}\right)\cdot(-1) + \arctan\left(\frac{1}{u}\right)\\ &= \arctan\left(-\frac{1}{u}\right) + \arctan\left(\frac{1}{u}\right) \\ &= 0 \end{align*}