Show that $$\int_{0}^{1} \ln\left(\frac{1-e^{-tu}}{t}\right) \mathrm{d}u \underset{t \rightarrow 0^+}{\longrightarrow} -1.$$ One may begin by establishing that $x \mapsto \frac{1-e^{-x}}{x}$ is decreasing on $\mathbf{R}_+$.
Show that
$$\int_{0}^{1} \ln\left(\frac{1-e^{-tu}}{t}\right) \mathrm{d}u \underset{t \rightarrow 0^+}{\longrightarrow} -1.$$
One may begin by establishing that $x \mapsto \frac{1-e^{-x}}{x}$ is decreasing on $\mathbf{R}_+$.