int_{-pi}^{pi} frac{sin^4 x}{sin^4 x + cos^4 | vedclass

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(D) माना $I = \int_{-\pi}^{\pi} \frac{\sin^4 x}{\sin^4 x + \cos^4 x} \, dx$ है। चूंकि फलन $f(x) = \frac{\sin^4 x}{\sin^4 x + \cos^4 x}$ एक सम फलन है,इ

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$\pi$

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(D) माना $I = \int_{-\pi}^{\pi} \frac{\sin^4 x}{\sin^4 x + \cos^4 x} \, dx$ है। चूंकि फलन $f(x) = \frac{\sin^4 x}{\sin^4 x + \cos^4 x}$ एक सम फलन है,इसलिए $I = 2 \int_{0}^{\pi} \frac{\sin^4 x}{\sin^4 x + \cos^4 x} \, dx$ होगा। गुणधर्म $\int_{0}^{2a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx$ का उपयोग करते हुए,जहाँ $f(\pi - x) = f(x)$ है,$I = 2 	imes 2 \int_{0}^{\pi/2} \frac{\sin^4 x}{\sin^4 x + \cos^4 x} \, dx = 4 \int_{0}^{\pi/2} \frac{\sin^4 x}{\sin^4 x + \cos^4 x} \, dx$ ..... $(i)$। गुणधर्म $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$ का उपयोग करते हुए,$I = 4 \int_{0}^{\pi/2} \frac{\cos^4 x}{\cos^4 x + \sin^4 x} \, dx$ ..... $(ii)$। $(i)$ और $(ii)$ को जोड़ने पर,$2I = 4 \int_{0}^{\pi/2} \frac{\sin^4 x + \cos^4 x}{\sin^4 x + \cos^4 x} \, dx = 4 \int_{0}^{\pi/2} 1 \, dx = 4 	imes \frac{\pi}{2} = 2\pi$। अतः,$I = \pi$।

$\int_{-\pi}^{\pi} \frac{\sin^4 x}{\sin^4 x + \cos^4 x} \, dx = $

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