The function $h$ is defined on $\mathbb{R}$ by
$$h(u) = u - [u] - 1/2$$
Let $\varphi$ be the application defined for all $x > 0$ by:
$$\varphi(x) = \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$
By repeating the integration by parts from question IV.C, prove that the application $\varphi$ is of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{*}$ and that for all $x > 0$,
$$\varphi^{\prime}(x) = -\int_{0}^{+\infty} \frac{h(u)}{(u+x)^{2}} du$$