int_{-frac{pi}{2}}^{frac{pi}{2}} frac{sin ^2 | vedclass

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(A) माना $I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin ^2 x}{1+2^x} \,d x$ ... $(i)$ गुणधर्म $\int_{a}^{b} f(x) \,d x = \int_{a}^{b} f(a+b-x)

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$\frac{\pi}{4}$

Vedclass · Answer

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

$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin ^2 x}{1+2^x} \,d x$ का मान है

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