Question Number 53385 by gunawan last updated on 21/Jan/19
$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integral}\:\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:\frac{{x}\:{dx}}{\mathrm{1}+\mathrm{cos}\:\alpha\:\mathrm{sin}\:\alpha}\:, \\ $$$$\mathrm{0}<\:\alpha<\pi\:\:\:\mathrm{is} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 21/Jan/19
$$\frac{\mathrm{1}}{\mathrm{1}+{sin}\alpha{cos}\alpha}×\mid\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\mid_{\mathrm{0}} ^{\pi} \\ $$$$=\frac{\pi^{\mathrm{2}} }{\mathrm{2}\left(\mathrm{1}+{sin}\alpha{cos}\alpha\right)} \\ $$