The Integral $\int \frac{\sin^{2} x \cos^{2} x}{(\sin^{5} x + \cos^{3} x \sin^{2} x + \sin^{3} x \cos^{2} x + c o s^{5} x)^{2}} d x$ is equal to
(where C is a constant of integration)

$\frac{- 1}{1 + c o t^{3} x} + C$

$\frac{1}{3 (1 + \tan^{3} x)} + C$

$\frac{- 1}{3 (1 + \tan^{3} x)} + C$

$\frac{1}{1 + c o t^{3} x} + C$

Solution

$\frac{- 1}{3 (1 + \tan^{3} x)} + C$

$I = \int \frac{\sin^{2} x . \cos^{2} x d x}{{(\sin^{2} x + \cos^{2} x) (\sin^{3} x + \cos^{3} x)}^{2}} D i v i d e n u m b e r a t o r a n d d e n o m i n a t o r b y \cos^{6} x \Rightarrow I = \int \frac{\tan^{2} x s e c^{2} x d x}{(1 + \tan^{3} x)^{2}} L e t \tan^{3} x = z \Rightarrow 3 \tan^{2} x . s e c^{2} x d x = d z 1 = \frac{1}{3} \int \frac{d z}{z^{2}} = \frac{- 1}{3 z} + C = \frac{- 1}{3 (1 + \tan^{3} x)} + C$

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