Question Number 38122 by maxmathsup by imad last updated on 22/Jun/18
$${calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{sinx}}{\:\sqrt{\mathrm{1}+{cos}^{\mathrm{2}} {x}}}{dx} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jun/18
$${t}={cosx}\:\:\:{dt}=−{sinxdx} \\ $$$$\int_{\mathrm{1}} ^{−\mathrm{1}} \frac{−{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$$$\int_{−\mathrm{1}} ^{\mathrm{1}} \frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$$${l} \\ $$$$=\mid{ln}\left({t}+\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\right)\mid_{−\mathrm{1}} ^{\mathrm{1}} \\ $$$$={ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\:\right)−{ln}\left(−\mathrm{1}+\sqrt{\mathrm{2}}\:\right) \\ $$