Let I=int_{-frac{pi}{4}}^{frac{pi}{4}} frac{1 | vedclass

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(A) Let $I=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{2-\cos 2 x}\left(\frac{3}{\pi}+\log \left(\frac{4+\sin x}{4-\sin x}\right)\right) d x$. Spli

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$4$

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(A) Let $I=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{2-\cos 2 x}\left(\frac{3}{\pi}+\log \left(\frac{4+\sin x}{4-\sin x}ight)ight) d x$. Split the integral into two parts: $I=\frac{3}{\pi} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{2-\cos 2 x} d x + \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{2-\cos 2 x} \log \left(\frac{4+\sin x}{4-\sin x}ight) d x$. The second integral is zero because the integrand is an odd function. Thus,$I=\frac{3}{\pi} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{2-\cos 2 x} d x = \frac{6}{\pi} \int_0^{\frac{\pi}{4}} \frac{1}{2-\cos 2 x} d x$. Using $\cos 2x = \frac{1-	an^2 x}{1+	an^2 x}$,we get: $I=\frac{6}{\pi} \int_0^{\frac{\pi}{4}} \frac{\sec^2 x}{2(1+	an^2 x) - (1-	an^2 x)} d x = \frac{6}{\pi} \int_0^{\frac{\pi}{4}} \frac{\sec^2 x}{1+3	an^2 x} d x$. Let $	an x = t$,then $\sec^2 x d x = d t$. As $x: 0 	o \frac{\pi}{4}$,$t: 0 	o 1$. $I=\frac{6}{\pi} \int_0^1 \frac{d t}{1+3t^2} = \frac{6}{\pi} \left[ \frac{1}{\sqrt{3}} 	an^{-1}(\sqrt{3}t) ight]_0^1 = \frac{6}{\pi \sqrt{3}} 	an^{-1}(\sqrt{3}) = \frac{2\sqrt{3}}{\pi} \cdot \frac{\pi}{3} = \frac{2}{\sqrt{3}}$. Therefore,$3I^2 = 3 \left( \frac{2}{\sqrt{3}} ight)^2 = 3 \cdot \frac{4}{3} = 4$.

Let $I=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{2-\cos 2 x}\left(\frac{3}{\pi}+\log \left(\frac{4+\sin x}{4-\sin x}\right)\right) d x$. Given that $\int \frac{d x}{1+k x^2}=\frac{1}{\sqrt{k}} \tan ^{-1}(\sqrt{k} x)+c, \tan ^{-1}(0)=0$ and $\tan ^{-1}(\sqrt{3})=\frac{\pi}{3}$. Then $3 I^2=$

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