int_{0}^{frac{pi}{2}} frac{sin^{frac{2}{3}} x | vedclass

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(A) Let $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx \quad ...(1)$ Using the property $\i

Vedclass · Accepted Answer

$\frac{\pi}{4}$

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(A) Let $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx \quad ...(1)$ Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$,we get: $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}}(\frac{\pi}{2}-x)}{\sin^{\frac{2}{3}}(\frac{\pi}{2}-x) + \cos^{\frac{2}{3}}(\frac{\pi}{2}-x)} dx$ Since $\sin(\frac{\pi}{2}-x) = \cos x$ and $\cos(\frac{\pi}{2}-x) = \sin x$,we have: $I = \int_{0}^{\frac{\pi}{2}} \frac{\cos^{\frac{2}{3}} x}{\cos^{\frac{2}{3}} x + \sin^{\frac{2}{3}} x} dx \quad ...(2)$ Adding equations $(1)$ and $(2)$: $2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx$ $2I = \int_{0}^{\frac{\pi}{2}} 1 dx$ $2I = [x]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2}$ $I = \frac{\pi}{4}$

$\int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx =$

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