Evaluate the following definite integral: int | vedclass

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(C) Let $I = \int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} d x \quad \dots(1)$Using the property $\

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$\frac{\pi}{4}$

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(C) Let $I = \int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} d x \quad \dots(1)$Using the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$, we get:$I = \int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}}(\frac{\pi}{2}-x)}{\sin ^{\frac{3}{2}}(\frac{\pi}{2}-x)+\cos ^{\frac{3}{2}}(\frac{\pi}{2}-x)} d x$Since $\sin(\frac{\pi}{2}-x) = \cos x$ and $\cos(\frac{\pi}{2}-x) = \sin x$, we have:$I = \int_0^{\frac{\pi}{2}} \frac{\cos ^{\frac{3}{2}} x}{\cos ^{\frac{3}{2}} x+\sin ^{\frac{3}{2}} x} d x \quad \dots(2)$Adding $(1)$ and $(2)$:$2I = \int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x + \cos ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x + \cos ^{\frac{3}{2}} x} d x$$2I = \int_0^{\frac{\pi}{2}} 1 d x$$2I = [x]_0^{\frac{\pi}{2}} = \frac{\pi}{2}$$I = \frac{\pi}{4}$

Evaluate the following definite integral: $\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} d x$

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