Integration Questions

Question Number 122979 by bemath last updated on 21/Nov/20

$$\:\:\int_{\mathrm{0}} ^{\pi} \frac{{x}\:\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{3}+\mathrm{sin}\:^{\mathrm{2}} {x}}}\:{dx}\:? \\$$

Answered by liberty last updated on 21/Nov/20

$$\psi\left({x}\right)=\int_{\mathrm{0}} ^{\pi} \frac{{x}\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{4}−\mathrm{cos}\:^{\mathrm{2}} {x}}}\:{dx} \\$$$$\:{replacing}\:{x}\:{by}\:\pi−{x}\:,\:{give}\: \\$$$$\psi\left({x}\right)=\int_{\pi} ^{\mathrm{0}} \frac{\left(\pi−{x}\right)\mathrm{sin}\:\left(\pi−{x}\right)}{\:\sqrt{\mathrm{4}−\mathrm{cos}\:^{\mathrm{2}} \left(\pi−{x}\right)}}\:\left(−{dx}\right) \\$$$$\psi\left({x}\right)=\int_{\mathrm{0}} ^{\pi} \frac{\left(\pi−{x}\right)\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{4}−\mathrm{cos}\:^{\mathrm{2}} {x}}}\:{dx}\: \\$$$${adding}\:{the}\:{both}\:{integral}\:,\:{we}\:{obtain} \\$$$$\mathrm{2}\psi\left({x}\right)=\int_{\mathrm{0}} ^{\pi} \frac{\pi\mathrm{sin}\:{x}}{\:\sqrt{\left(\mathrm{2}\right)^{\mathrm{2}} −\mathrm{cos}\:^{\mathrm{2}} \left({x}\right)}}\:{dx} \\$$$$\psi\left({x}\right)=−\frac{\pi}{\mathrm{2}}\left[\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{cos}\:{x}}{\mathrm{2}}\right)\right]_{\mathrm{0}} ^{\pi} \\$$$$\psi\left({x}\right)=−\frac{\pi}{\mathrm{2}}\left(−\frac{\pi}{\mathrm{3}}\right)=\frac{\pi}{\mathrm{6}}.\:\blacktriangle \\$$$$\\$$