Answer:
Option B
Explanation:
We have
I= $\int_{}^{} \frac{\sin^{2}x cos^{2}x}{(\sin^{5}x+\cos^{3}x\sin^{2}x+\sin^{3}x\cos^{2}x+\cos^{5}x)^{2}}dx$
$\int_{}^{}\frac{\sin^{2}x cos^{2}x}{( \sin^{3}x( \sin^{2}x+\cos^{2}x)+\cos^{3}x( \sin^{2}x+\cos^{2}x))^{2}}dx$
= $\int_{}^{}\frac{\sin^{2}x cos^{2}x}{( \sin^{3}x+\cos^{3}x)^{2}}dx$
= $\int_{}^{}\frac{\sin^{2}x cos^{2}x}{\cos^{6}x( 1+\tan^{3}x)^{2}}dx$
= $\int_{}^{} \frac{\tan^{2}x\sec^{2}x}{( 1+\tan^{3}x)^{2}}dx$
Put $\tan^{2}x=1\Rightarrow 3\tan^{3}x \sec^{2}x dx=dt$
$\therefore $ $I=\frac{1}{3}\int_{}^{} \frac{dt}{( 1+t)^{2}}$
$\Rightarrow$ I= $\frac{-1}{3( 1+t)}+C$
$\Rightarrow$ $I= \frac{-1}{3( 1+\tan^{3}x)}+C$
