Let $g$ be the function defined by
$$\begin{aligned}
g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\
\theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t
\end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Deduce that:
$$\forall \theta \in ] 0 ; \pi \left[ , \quad g ( \theta ) \sin ( \theta x ) = \int _ { \cot ( \theta ) } ^ { + \infty } \frac { ( u \sin ( \theta ) - \cos ( \theta ) ) ^ { x } } { 1 + u ^ { 2 } } \mathrm {~d} u , \right.$$
where $\cot ( \theta ) = \frac { \cos ( \theta ) } { \sin ( \theta ) }$.