Integration Questions

Question Number 175351 by cortano1 last updated on 28/Aug/22

$$\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{sin}\:^{\mathrm{3}} {x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}\:{dx}\:=? \\$$

Answered by som(math1967) last updated on 28/Aug/22

$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sin}^{\mathrm{3}} \left(\frac{\pi}{\mathrm{2}}+\mathrm{0}−{x}\right){dx}}{{sin}\left(\frac{\pi}{\mathrm{2}}+\mathrm{0}−{x}\right)+{cos}\left(\frac{\pi}{\mathrm{2}}+\mathrm{0}−{x}\right)} \\$$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{cos}^{\mathrm{3}} {xdx}}{{cosx}+{sinx}} \\$$$$\mathrm{2}{I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sin}^{\mathrm{3}} {x}+{cos}^{\mathrm{3}} {x}}{{sinx}+{cosx}}{dx} \\$$$$\mathrm{2}{I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sin}^{\mathrm{2}} {x}−{sinxcox}+{cos}^{\mathrm{2}} {x}\right){dx} \\$$$$\mathrm{2}{I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {dx}−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}\mathrm{2}{xdx} \\$$$$\mathrm{2}{I}=\left[{x}+\frac{{cos}\mathrm{2}{x}}{\mathrm{4}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\$$$$\mathrm{2}{I}=\left(\frac{\pi}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{4}}\right)−\left(\mathrm{0}+\frac{\mathrm{1}}{\mathrm{4}}\right) \\$$$${I}=\frac{\pi}{\mathrm{4}}\:−\frac{\mathrm{1}}{\mathrm{4}} \\$$

Commented by cortano1 last updated on 28/Aug/22

$${by}\:{King}\:{Formula} \\$$

Commented by som(math1967) last updated on 28/Aug/22

$${yes} \\$$