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

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Question

(A) माना $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx \quad ...(1)$ गुणधर्म $\int_{0}^{a}

Vedclass · Accepted Answer

$\frac{\pi}{4}$

Vedclass · Answer

(A) माना $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx \quad ...(1)$ गुणधर्म $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$ का उपयोग करने पर,हमें प्राप्त होता है: $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}}(\frac{\pi}{2}-x)}{\sin^{\frac{2}{3}}(\frac{\pi}{2}-x) + \cos^{\frac{2}{3}}(\frac{\pi}{2}-x)} dx$ चूंकि $\sin(\frac{\pi}{2}-x) = \cos x$ और $\cos(\frac{\pi}{2}-x) = \sin x$,इसलिए: $I = \int_{0}^{\frac{\pi}{2}} \frac{\cos^{\frac{2}{3}} x}{\cos^{\frac{2}{3}} x + \sin^{\frac{2}{3}} x} dx \quad ...(2)$ समीकरण $(1)$ और $(2)$ को जोड़ने पर: $2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx$ $2I = \int_{0}^{\frac{\pi}{2}} 1 dx$ $2I = [x]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2}$ $I = \frac{\pi}{4}$

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

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