Integration Questions

Question Number 7300 by Tawakalitu. last updated on 22/Aug/16

$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \:\:\:\frac{{sinx}}{{sinx}\:+\:{cosx}}\:{dx}\: \\$$

Answered by Yozzia last updated on 22/Aug/16

$${I}=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{sinx}}{{sinx}+{cosx}}{dx} \\$$$${Q}=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{cosx}}{{sinx}+{cosx}}{dx} \\$$$${I}+{Q}=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{sinx}+{cosx}}{{sinx}+{cosx}}{dx}=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {dx}=\frac{\pi}{\mathrm{2}} \\$$$${Q}−{I}=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{cosx}−{sinx}}{{sinx}+{cosx}}{dx}=\int_{\mathrm{1}} ^{\mathrm{1}} \frac{\mathrm{1}}{{u}}{du}=\mathrm{0} \\$$$$\Rightarrow{I}={Q}\:\therefore\:\mathrm{2}{I}=\frac{\pi}{\mathrm{2}}\Rightarrow{I}=\frac{\pi}{\mathrm{4}}. \\$$$$\\$$

Commented by Yozzia last updated on 22/Aug/16

$${I}\left({m}\right)=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{sin}^{{m}} {x}}{{cos}^{{m}} {x}+{sin}^{{m}} {x}}{dx}\:\:\:{m}\in\mathbb{R}. \\$$$$\because\:\int_{\mathrm{0}} ^{{a}} {f}\left({x}\right){dx}=\int_{\mathrm{0}} ^{{a}} {f}\left({a}−{x}\right){dx} \\$$$$\Rightarrow{I}\left({m}\right)=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{sin}^{{m}} \left(\frac{\pi}{\mathrm{2}}−{x}\right)}{{cos}^{{m}} \left(\frac{\pi}{\mathrm{2}}−{x}\right)+{sin}^{{m}} \left(\frac{\pi}{\mathrm{2}}−{x}\right)}{dx} \\$$$${Since}\:{sin}\left(\mathrm{0}.\mathrm{5}\pi−{x}\right)={cosx}\:{and}\:{cos}\left(\mathrm{0}.\mathrm{5}\pi−{x}\right)={sinx} \\$$$$\Rightarrow{I}\left({m}\right)=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{cos}^{{m}} {x}}{{sin}^{{m}} {x}+{cos}^{{m}} {x}}{dx}. \\$$$$\therefore\:\mathrm{2}{I}\left({m}\right)=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{sin}^{{m}} {x}+{cos}^{{m}} {x}}{{sin}^{{m}} {x}+{cos}^{{m}} {x}}{dx}=\frac{\pi}{\mathrm{2}} \\$$$$\Rightarrow{I}\left({m}\right)=\frac{\pi}{\mathrm{4}}\:{or}\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{sin}^{{m}} {x}}{{cos}^{{m}} {x}+{sin}^{{m}} {x}}{dx}=\frac{\pi}{\mathrm{4}}. \\$$$$\\$$

Commented by Tawakalitu. last updated on 22/Aug/16

$${Wow},\:{thanks}\:{so}\:{much}.\:{i}\:{really}\:{appreciate} \\$$