Calculate and student this integral :

$\displaystyle \int_0^{ + \infty } {\left( {\frac{{\arctan \left( {\frac{1}{x}} \right)}}{{x^2 + 1}}} \right)} dx

$

Same question :

$\displaystyle \int_0^{ + \infty } {\left( {\sin \left( {\frac{1}{{x^\alpha }}} \right)} \right)} dx.....\left( {\alpha \in \Re } \right)$