A) \[A=\frac{1}{4},B=-\frac{1}{3},C=\frac{1}{8},D\in R\]
B) \[A=\frac{1}{8},B=\frac{1}{4},C=\frac{1}{3},D\in R\]
C) \[A=0,B=-\frac{1}{6},C=\frac{1}{8},D\in R\]
D) None of these
Correct Answer: A
Solution :
[a] \[I=\int{{{\sin }^{3}}x.{{\cos }^{5}}xdx}\] Put \[\sin \,\,x=t\Rightarrow cos\,\,x\,\,dx=dt\] \[I=\int{{{\sin }^{3}}x.{{\cos }^{4}}x.\cos x\,\,dx=\int{{{t}^{3}}{{\left( 1-{{t}^{2}} \right)}^{2}}dt}}\] \[=\int{\left( {{t}^{3}}-2{{t}^{5}}+{{t}^{7}} \right)dt=\frac{1}{4}{{t}^{4}}-\frac{2}{6}{{t}^{6}}+\frac{1}{8}{{t}^{8}}+D}\] \[=\frac{1}{4}{{\sin }^{4}}x-\frac{1}{3}{{\sin }^{6}}x+\frac{1}{8}{{\sin }^{8}}x+D\]
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