Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Deduce that $\forall x \in \mathbb { R } \backslash \{ - 1,1 \}$
$$f ^ { \prime } ( x ) = 4 \int _ { 0 } ^ { + \infty } \frac { ( x + 1 ) t ^ { 2 } + ( x - 1 ) } { \left( ( x + 1 ) ^ { 2 } t ^ { 2 } + ( x - 1 ) ^ { 2 } \right) \left( t ^ { 2 } + 1 \right) } \mathrm { d } t$$