Question Number 30216 by abdo imad last updated on 18/Feb/18
$${let}\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dt}}{{x}^{\mathrm{2}} \:+{cos}^{\mathrm{2}} {t}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{I}\left({x}\right)=\:\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{{x}^{\mathrm{2}} \:+{cos}^{\mathrm{2}} {t}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I}\left({x}\right). \\ $$