Question Number 30179 by abdo imad last updated on 17/Feb/18
$${find}\:\:\int\:\:\:\:\frac{{dt}}{\mathrm{1}+{cost}\:+{sint}}\:\:. \\ $$
Commented by abdo imad last updated on 24/Feb/18
$${the}\:{ch}.\:{tan}\left(\frac{{t}}{\mathrm{2}}\right)={x}\:{give}\: \\ $$$${I}\:=\:\int\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}−{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }}\:\frac{\mathrm{2}{dx}}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$=\:\int\:\:\:\:\:\:\frac{\mathrm{2}{dx}}{\mathrm{1}+{x}^{\mathrm{2}} \:+\mathrm{1}−{x}^{\mathrm{2}} \:+\mathrm{2}{x}}\:=\:\int\:\:\:\:\frac{\mathrm{2}{dx}}{\mathrm{2}+\mathrm{2}{x}}\:=\:\int\:\:\:\frac{{dx}}{\mathrm{1}+{x}} \\ $$$$={ln}\mid\mathrm{1}+{x}\mid\:+\lambda=\:\:{ln}\mid\mathrm{1}\:+{tan}\left(\frac{{t}}{\mathrm{2}}\right)\mid\:+\lambda\:. \\ $$