Find 2 x 2 x d x 9- ^ 4 (2 x) - Vidyadip

Question

Find $\int \frac{\sin 2 x \cos 2 x d x}{\sqrt{9-\cos ^{4}(2 x)}}$

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Answer

Let $I=\int \frac{\sin 2 x \cos 2 x}{\sqrt{9-\cos ^{4} 2 x}} d x$
Put $\cos^2$ (2x) = t so that 2 sin 2x cos 2x dx = -dt
Therefore,
I = $-\frac{1}{2} \int \frac{d t}{\sqrt{9-t^{2}}}$ = $-\frac{1}{2} \sin ^{-1}\left(\frac{t}{3}\right)+\mathrm{C}=-\frac{1}{2} \sin ^{-1}\left[\frac{1}{3} \cos ^{2} 2 x\right]+\mathrm{C}$

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