First slide

Methods of integration

Question

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

Easy

A

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

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

$4 \cos^{- 1} \cos^{4} 2 x + c$
D

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

Solution

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

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