Integration Questions

Question Number 90997 by jagoll last updated on 27/Apr/20

$$\int\underset{\mathrm{0}} {\overset{\pi} {\:}}\:\frac{\mathrm{sin}\:{x}\:{dx}}{\mathrm{1}+\mathrm{sin}\:{x}} \\$$

Commented byjohn santu last updated on 27/Apr/20

Commented byjagoll last updated on 27/Apr/20

$${thank}\:{you} \\$$

Commented bymathmax by abdo last updated on 27/Apr/20

$${I}\:=\int_{\mathrm{0}} ^{\pi} \:\frac{{sinx}}{\mathrm{1}+{sinx}}{dx}\:\Rightarrow\:{I}\:=\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{1}+{sinx}−\mathrm{1}}{\mathrm{1}+{sinx}}{dx} \\$$ $$=\pi\:−\int_{\mathrm{0}} ^{\pi} \:\frac{{dx}}{\mathrm{1}+{sinx}}\:\:{we}\:{have}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dx}}{\mathrm{1}+{sinx}}\:=_{{tan}\left(\frac{{x}}{\mathrm{2}}\right)={t}} \:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{2}{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\left(\mathrm{1}+\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}} }\right)} \\$$ $$\left.=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{2}{dt}}{\mathrm{1}+{t}^{\mathrm{2}} \:+\mathrm{2}{t}}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{2}{dt}}{\left({t}+\mathrm{1}\right)^{\mathrm{2}} }\:=−\frac{\mathrm{2}}{{t}+\mathrm{1}}\right]_{\mathrm{0}} ^{\infty} \:=\mathrm{2}\:\Rightarrow{I}\:=\pi−\mathrm{2} \\$$