sin2x cos2x / root (9-cos^4 2x) integrate

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Solution

Dear student,

$Let I = \int \frac{\cos 2 x . \sin 2 x}{\sqrt{9 - \cos^{4} 2 x}} dx Take u = \frac{\cos^{2} 2 x}{3} \frac{du}{dx} = - \frac{4 \cos 2 x . \sin 2 x}{3} \Rightarrow (\cos 2 x . \sin 2 x) dx = - \frac{3}{4} du I = - \frac{3}{4} \int \frac{du}{\sqrt{9 - 9 u^{2}}} = \frac{- 1}{4} \int \frac{du}{\sqrt{1 - u^{2}}} = \frac{- 1}{4} \sin^{- 1} u + C = \frac{- 1}{4} \sin^{- 1} (\frac{\cos^{2} 2 x}{3}) + C$

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