Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Show that $f$ is continuous on $\mathbb { R }$ and that $f ( 1 ) = f ( - 1 ) = 0$.
One may show that $\forall x \in \mathbb { R } , x ^ { 2 } - 2 x \cos \theta + 1 \geqslant \sin ^ { 2 } \theta$ and use the dominated convergence theorem.