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Q.
The integral $\int \frac{ \sin^{2 }x \cos^{2}x }{\left(\sin^{5} x + \cos^{3} x \sin^{2} x + \sin^{3} x \cos^{2} x + \cos^{5} x\right)^{2}} dx $ is equal to
$I=\int \frac{\sin ^{2} x \cdot \cos ^{2} x d x}{\left\{\left(\sin ^{2} x+\cos ^{2} x\right)\left(\sin ^{3} x+\cos ^{3} x\right)\right\}^{2}}$
Dividing the numerator and denominator by $\cos ^{6} x$
$\Rightarrow I=\int \frac{\tan ^{2} x \sec ^{2} x d x}{\left(1+\tan ^{3} x\right)^{2}}$
Let, $\tan ^{3} x=z$
$\Rightarrow 3 \tan ^{2} x \cdot \sec ^{2} x d x=d z$
$I=\frac{1}{3} \int \frac{d z}{z^{2}}=\frac{-1}{3 z}+C$
$=\frac{-1}{3\left(1+\tan ^{3} x\right)}+C$