Question Number 123023 by mnjuly1970 last updated on 21/Nov/20
$$\:\:\:\:\:\:\:\:\:\:…\:{advanced}\:\:{calculus}… \\ $$$${calculate}::: \\ $$$$\:\:\:\:\:\mathrm{I}:\overset{???} {=}\:\int_{\mathrm{0}} ^{\:\pi} \frac{{x}}{\mathrm{1}−{sin}\left({x}\right){cos}\left({x}\right)}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:………………………….. \\ $$
Answered by mnjuly1970 last updated on 24/Nov/20
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{evaluation}:\:\:\mathrm{I}\overset{?} {=}\int_{\mathrm{0}} ^{\:\pi} \frac{{x}}{\mathrm{1}−{sin}\left({x}\right){cos}\left({x}\right)}{dx} \\ $$$$\:\:\:{solution}: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{I}=\left[\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}}{\mathrm{1}−{sin}\left({x}\right){cos}\left({x}\right)}{dx}=\mathrm{I}_{\mathrm{1}} \right]+\left[\int_{\frac{\pi}{\mathrm{2}}\:} ^{\:\pi} \frac{{x}}{\mathrm{1}−{xsin}\left({x}\right){cos}\left({x}\right)}{dx}=\mathrm{I}_{\mathrm{2}} \right] \\ $$$$\:\:\:\:\:\:\:\mathrm{I}_{\mathrm{1}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{2}{x}}{\mathrm{2}−{sin}\left(\mathrm{2}{x}\right)}{dx}=\pi\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}}{\mathrm{2}−{sin}\left(\mathrm{2}{x}\right)}{dx}−\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}}{\mathrm{2}−{sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$$$\:\:\:\:\:\mathrm{2I}_{\mathrm{1}} =\pi\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{dx}}{\mathrm{2}−\frac{\mathrm{2}{tan}\left({x}\right)}{\mathrm{1}+{tan}^{\mathrm{2}} \left({x}\right)}}=\pi\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}+{tan}^{\mathrm{2}} \left({x}\right)}{\mathrm{2}{tan}^{\mathrm{2}} \left({x}\right)−\mathrm{2}{tan}\left({x}\right)+\mathrm{2}}{dx} \\ $$$$\:\:\overset{{tan}\left({x}\right)={y}} {=}\frac{\pi}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \frac{{dy}}{{y}^{\mathrm{2}} −{y}+\mathrm{1}}=\frac{\pi}{\mathrm{2}}\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\left[{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\left({y}−\frac{\mathrm{1}}{\mathrm{2}}\right)\right)\overset{\infty} {\right]}_{\mathrm{0}} \\ $$$$\:\:\:\:\:\:=\frac{\pi}{\:\sqrt{\mathrm{3}}}\left[\frac{\pi}{\mathrm{2}}+\frac{\pi}{\mathrm{6}}\right]=\frac{\mathrm{2}\pi^{\mathrm{2}} }{\:\mathrm{3}\sqrt{\mathrm{3}}}\:\Rightarrow\:\mathrm{I}_{\mathrm{1}} =\frac{\pi^{\mathrm{2}} }{\mathrm{3}\sqrt{\mathrm{3}}}\:\checkmark\: \\ $$$$\:\:\:\:\:\mathrm{I}_{\mathrm{2}} =\int_{\frac{\pi}{\mathrm{2}}} ^{\:\pi} \frac{{x}}{\mathrm{1}−{sin}\left({x}\right){cos}\left({x}\right)}{dx} \\ $$$$\:\:\:\:\:=\frac{\mathrm{3}\pi}{\mathrm{2}}\int_{\frac{\pi}{\mathrm{2}}} ^{\:\pi} \frac{\mathrm{2}}{\mathrm{2}−{sin}\left(\mathrm{2}{x}\right)}{dx}−\mathrm{I}_{\mathrm{2}} \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{2I}_{\mathrm{2}} \underset{\mathrm{I}_{\mathrm{1}} } {\overset{{similarly}} {=}}\left(\frac{\mathrm{3}\pi}{\mathrm{2}}\left(\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\right)\left[{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\left({t}−\frac{\mathrm{1}}{\mathrm{2}}\right)\right]_{−\infty} ^{\mathrm{0}} \right.\right. \\ $$$$\:\:\:\:\:\mathrm{2I}_{\mathrm{2}} =\frac{\mathrm{3}\pi}{\:\sqrt{\mathrm{3}}}\left[\frac{−\pi}{\mathrm{6}}+\frac{\pi}{\mathrm{2}}\right]=\frac{\mathrm{3}\pi^{\mathrm{2}} }{\mathrm{3}\sqrt{\mathrm{3}}}\:\Rightarrow\:\mathrm{I}_{\mathrm{2}} =\frac{\pi^{\mathrm{2}} }{\:\mathrm{2}\sqrt{\mathrm{3}}} \\ $$$$\:\:\:\:\:\therefore\:\mathrm{I}=\mathrm{I}_{\mathrm{1}} +\mathrm{I}_{\mathrm{2}} =\frac{\pi^{\mathrm{2}} }{\:\sqrt{\mathrm{3}}}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}\right)=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{6}\sqrt{\mathrm{3}}}\:\:\checkmark\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$$${r} \\ $$