A) \[\frac{1}{1+{{\cot }^{3}}x}+C\]
B) \[\frac{-1}{1+{{\cot }^{3}}x}+C\]
C) \[\frac{1}{3(1+{{\tan }^{3}}x)}+C\]
D) \[\frac{-1}{3(1+{{\tan }^{3}}x)}+C\] (where C is a constant of integration)
Correct Answer: D
Solution :
\[\int_{{}}^{{}}{\frac{{{\sin }^{2}}x.{{\cos }^{2}}x}{{{({{\sin }^{5}}x+{{\cos }^{3}}x-{{\sin }^{2}}x-{{\sin }^{3}}x.{{\cos }^{2}}x+{{\cos }^{5}}x)}^{2}}}dx}\] \[\int_{{}}^{{}}{\frac{{{\sin }^{2}}x.{{\cos }^{2}}x}{{{(({{\sin }^{3}}x+{{\cos }^{3}}x)(si{{n}^{2}}x+{{\cos }^{2}}x))}^{2}}}dx}\] \[\int_{{}}^{{}}{\frac{{{\tan }^{2}}x.{{\sec }^{2}}x}{{{({{\tan }^{3}}x+1)}^{2}}}}\] Put \[{{\tan }^{3}}x+1=t\] \[\frac{1}{3}\int_{{}}^{{}}{\frac{dt}{{{t}^{2}}}=-\frac{1}{3t}+C\Rightarrow -\frac{1}{3({{\tan }^{3}}x+1)}+C}\]
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