Question Number 55271 by maxmathsup by imad last updated on 20/Feb/19
$$\left.\mathrm{1}\right)\:{let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{cost}}{\mathrm{3}\:+{sin}\left({xt}\right)}{dt} \\ $$$${find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{tcos}\left({xt}\right){cost}}{\left(\mathrm{3}\:+{sin}\left({xt}\right)\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cost}}{\mathrm{3}+{sint}}\:\:{and}\:\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{t}\:{cos}^{\mathrm{2}} {t}}{\left(\mathrm{3}+{sint}\right)^{\mathrm{2}} }{dt} \\ $$