The value of int_{0}^{pi /2} frac{sin^{2/3} x | vedclass

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(A) Let $I = \int_{0}^{\pi /2} \frac{\sin^{2/3} x}{\sin^{2/3} x + \cos^{2/3} x} dx$  $(1)$Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-

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$\pi /4$

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

The value of $\int_{0}^{\pi /2} \frac{\sin^{2/3} x}{\sin^{2/3} x + \cos^{2/3} x} dx$ is

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