First slide

Methods of integration

Question

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

Moderate

A

$\frac{1}{2} \cos^{- 1} (\frac{1}{3} \cos^{2} 2 x) + c$
B

$\frac{- 1}{4} \cos^{- 1} (\frac{1}{3} \cos^{2} 2 x) + c$
C

$\frac{- 1}{2} \sin^{- 1} (\frac{1}{3} \cos^{2} 2 x) + c$
D

$\frac{- 1}{4} \sin^{- 1} (\frac{1}{3} \cos^{2} 2 x) + c$

Solution

$Put \cos^{2} 2 x = t$
$\Rightarrow - 2 \sin 2 x \cos 2 x 2 d x = d t \Rightarrow \sin 2 x \cos 2 x 2 d x = - \frac{1}{4} d t - \frac{1}{4} \int \frac{1}{\sqrt{9 - t^{4}}} d t = - \frac{1}{4} \sin^{- 1} (\frac{t}{3}) + c$

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