alpha 0 के लिए int_{-pi}^{pi} frac{cos^2 x} | vedclass

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(D) माना $I = \int_{-\pi}^{\pi} \frac{\cos^2 x}{1+\alpha^x} \, dx$ --- $(i)$गुणधर्म $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a+b-x) \, dx$ का उपयोग क

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$\frac{\pi}{2}$

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(D) माना $I = \int_{-\pi}^{\pi} \frac{\cos^2 x}{1+\alpha^x} \, dx$ --- $(i)$गुणधर्म $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a+b-x) \, dx$ का उपयोग करने पर:$I = \int_{-\pi}^{\pi} \frac{\cos^2(-x)}{1+\alpha^{-x}} \, dx = \int_{-\pi}^{\pi} \frac{\cos^2 x}{1+\frac{1}{\alpha^x}} \, dx = \int_{-\pi}^{\pi} \frac{\alpha^x \cos^2 x}{\alpha^x + 1} \, dx$ --- $(ii)$$(i)$ और $(ii)$ को जोड़ने पर:$2I = \int_{-\pi}^{\pi} \frac{\cos^2 x + \alpha^x \cos^2 x}{1+\alpha^x} \, dx = \int_{-\pi}^{\pi} \frac{\cos^2 x(1+\alpha^x)}{1+\alpha^x} \, dx = \int_{-\pi}^{\pi} \cos^2 x \, dx$चूँकि $\cos^2 x$ एक सम फलन है,$2I = 2 \int_{0}^{\pi} \cos^2 x \, dx$$I = \int_{0}^{\pi} \cos^2 x \, dx = \int_{0}^{\pi} \frac{1+\cos 2x}{2} \, dx = \left[ \frac{x}{2} + \frac{\sin 2x}{4} \right]_{0}^{\pi} = \frac{\pi}{2} + 0 - (0 + 0) = \frac{\pi}{2}$

$\alpha > 0$ के लिए $\int_{-\pi}^{\pi} \frac{\cos^2 x}{1+\alpha^x} \, dx$ का मान है

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