For any real $\alpha > 0$, consider the function $h _ { \alpha } : t \mapsto \ln \left( \frac { 1 - t ^ { 2 } } { \alpha ^ { 2 } + t ^ { 2 } } \right)$. Show that $h _ { \alpha }$ is a continuous decreasing integrable function on $[ 0,1 [$.
For any real $\alpha > 0$, consider the function $h _ { \alpha } : t \mapsto \ln \left( \frac { 1 - t ^ { 2 } } { \alpha ^ { 2 } + t ^ { 2 } } \right)$. Show that $h _ { \alpha }$ is a continuous decreasing integrable function on $[ 0,1 [$.