A) \[\frac{{{\cos }^{5}}x}{5}-\frac{{{\cos }^{3}}x}{3}+c\]
B) \[\frac{{{\cos }^{5}}x}{5}+\frac{{{\cos }^{3}}x}{3}+c\]
C) \[\frac{{{\sin }^{5}}x}{5}-\frac{{{\sin }^{3}}x}{3}+c\]
D) \[\frac{{{\sin }^{5}}x}{5}+\frac{{{\sin }^{3}}x}{3}+c\]
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{{{\sin }^{3}}x{{\cos }^{2}}x\,dx}=\int_{{}}^{{}}{(1-{{\cos }^{2}}x){{\cos }^{2}}x\,.\,\sin x\,dx}\] Put \[\cos x=t\Rightarrow -\sin x\,dx=dt,\] then it reduces to \[-\int_{{}}^{{}}{({{t}^{2}}-{{t}^{4}})dt}=\frac{{{t}^{5}}}{5}-\frac{{{t}^{3}}}{3}+c=\frac{{{(\cos x)}^{5}}}{5}-\frac{{{(\cos x)}^{3}}}{3}+c\].
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