First slide

Methods of integration

Question

$\int \frac{\sin x \cos x}{\sin^{4} x + \cos^{4} x} d x =$

Easy

A

$\tan^{- 1} (\sin x) + c$
B

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

$\frac{1}{2} \tan^{- 1} (\tan^{2} x) + c$
D

$\frac{1}{2} \tan^{- 1} (\cot^{2} x) + c$

Solution

$\int \frac{\sin x . \cos x}{\cos^{4} x (1 + \tan^{4} x)} d x$
$\int \frac{\tan x \sec^{2} x}{1 + \tan^{4} x} \tan^{2} x = t$

$= \frac{1}{2} \int \frac{1}{1 + t^{2}} d t$

$= \frac{1}{2} \tan^{- 1} t + C$

$= \frac{1}{2} \tan^{- 1} (\tan^{2} x) + C$

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