int_{0}^{frac{pi}{2}} frac{sin ^{4} x}{sin ^{ | vedclass

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Question

(B) माना $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} x}{\sin ^{4} x+\cos ^{4} x} d x$ ..... $(1)$गुणधर्म $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x)

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$\frac{\pi}{4}$

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(B) माना $I = \int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} x}{\sin ^{4} x+\cos ^{4} x} d x$ ..... $(1)$गुणधर्म $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$ का उपयोग करने पर:$I = \int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} (\frac{\pi}{2} - x)}{\sin ^{4} (\frac{\pi}{2} - x) + \cos ^{4} (\frac{\pi}{2} - x)} dx$चूंकि $\sin(\frac{\pi}{2} - x) = \cos x$ और $\cos(\frac{\pi}{2} - x) = \sin x$,इसलिए:$I = \int_{0}^{\frac{\pi}{2}} \frac{\cos ^{4} x}{\cos ^{4} x + \sin ^{4} x} dx$ ..... $(2)$$(1)$ और $(2)$ को जोड़ने पर:$2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} x + \cos ^{4} x}{\sin ^{4} x + \cos ^{4} x} dx$$2I = \int_{0}^{\frac{\pi}{2}} 1 dx = [x]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2}$अतः,$I = \frac{\pi}{4}$.

$\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{4} x}{\sin ^{4} x+\cos ^{4} x} d x$ का मान ज्ञात कीजिए।

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