A) \[-\frac{1}{5}{{\cos }^{5}}x+\frac{2}{7}{{\cos }^{7}}x-\frac{1}{9}{{\cos }^{9}}x+c\]
B) \[\frac{1}{5}{{\cos }^{5}}x+\frac{2}{7}{{\cos }^{7}}x-\frac{1}{9}{{\cos }^{9}}x+c\]
C) \[\frac{1}{5}{{\cos }^{5}}x+\frac{2}{7}{{\cos }^{7}}x+\frac{1}{9}{{\cos }^{9}}x+c\]
D) None of these
Correct Answer: A
Solution :
Put \[\cos x=t\Rightarrow -\sin x\,dx=dt,\] then \[\int_{{}}^{{}}{{{(1-{{\cos }^{2}}x)}^{2}}.{{\cos }^{4}}x\sin x\,dx}=-\int_{{}}^{{}}{{{(1-{{t}^{2}})}^{2}}.\,{{t}^{4}}dt}\] \[=-\frac{{{t}^{5}}}{5}+\frac{2}{7}{{t}^{7}}-\frac{1}{9}{{t}^{9}}+c=-\frac{{{\cos }^{5}}x}{5}+\frac{2}{7}{{\cos }^{7}}x-\frac{1}{9}{{\cos }^{9}}x+c\]. Aliter : By reduction formula.
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