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

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

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

$$I={\int }_0^{\frac{\pi }{2}}\frac{{sin}^3(\frac{\pi }{2}+0-x)dx}{sin(\frac{\pi }{2}+0-x)+cos(\frac{\pi }{2}+0-x)}$$$$={\int }_0^{\frac{\pi }{2}}\frac{{cos}^{3}xdx}{cosx+sinx}$$$$2I={\int }_0^{\frac{\pi }{2}}\frac{{sin}^{3}x+{cos}^{3}x}{sinx+cosx}dx$$$$2I={\int }_0^{\frac{\pi }{2}}({sin}^{2}x-sinxcox+{cos}^{2}x)dx$$$$2I={\int }_0^{\frac{\pi }{2}}dx-\frac{1}{2}{\int }_0^{\frac{\pi }{2}}sin2xdx$$$$2I={[x+\frac{cos2x}{4}]}_0^{\frac{\pi }{2}}$$$$2I=(\frac{\pi }{2}-\frac{1}{4})-(0+\frac{1}{4})$$$$I=\frac{\pi }{4}-\frac{1}{4}$$

Commented by cortano1 last updated on 28/Aug/22

$$byKingFormula$$

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

$$yes$$

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