If int frac{sin 2x , dx}{sin^4 x + cos^4 x} = | vedclass

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(C) Let $I = \int \frac{\sin 2x \, dx}{\sin^4 x + \cos^4 x}$.We know that $\sin 2x = 2 \sin x \cos x$,so $I = \int \frac{2 \sin x \cos x}{\sin^4 x + \

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$3$

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(C) Let $I = \int \frac{\sin 2x \, dx}{\sin^4 x + \cos^4 x}$.We know that $\sin 2x = 2 \sin x \cos x$,so $I = \int \frac{2 \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx$.Divide the numerator and denominator by $\cos^4 x$:$I = \int \frac{2 \sin x \cos x / \cos^4 x}{(\sin^4 x + \cos^4 x) / \cos^4 x} \, dx = \int \frac{2 	an x \sec^2 x}{1 + 	an^4 x} \, dx$.Let $t = 	an^2 x$,then $dt = 2 	an x \sec^2 x \, dx$.Substituting these into the integral:$I = \int \frac{dt}{1 + t^2} = 	an^{-1}(t) + c = 	an^{-1}(	an^2 x) + c$.Comparing this with $	an^{-1}(f(x)) + c$,we get $f(x) = 	an^2 x$.Therefore,$f\left(\frac{\pi}{3}ight) = 	an^2\left(\frac{\pi}{3}ight) = (\sqrt{3})^2 = 3$.

If $\int \frac{\sin 2x \, dx}{\sin^4 x + \cos^4 x} = \tan^{-1}(f(x)) + c$,then $f\left(\frac{\pi}{3}\right) = $

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