A) \[{{a}_{1}}\] and \[{{a}_{2}}\]
B) \[{{a}_{0}}\] and \[{{a}_{3}}\]
C) \[{{a}_{2}}\] and \[{{a}_{3}}\]
D) \[{{a}_{1}}\] and \[{{a}_{3}}\]
Correct Answer: D
Solution :
Let \[I=\int_{0}^{\pi }{\underset{r=0}{\mathop{\overset{3}{\mathop{\Sigma }}\,}}\,}{{a}_{r}}\,{{\cos }^{3-r}}\,x\,{{\sin }^{r}}\,x\,dx\] \[=\int_{0}^{\pi }{{{a}_{0}}\,{{\cos }^{3}}\,x\,dx+\int_{0}^{\pi }{{{a}_{1}}\,{{\cos }^{2}}\,x\,\,\sin \,x\,dx}}\] \[+\int_{0}^{\pi }{{{a}_{2}}}\,\cos \,x\,\,{{\sin }^{2}}x\,dx+\int_{0}^{\pi }{{{a}_{3}}\,{{\sin }^{3}}\,x\,\,dx}\] Since, \[\int_{0}^{2a}{f(x)\,dx=\left\{ \begin{matrix} 2\int_{0}^{a}{f(x)\,dx,\,if\,(2a-x)\,=\,f(x)} \\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,if\,\,f(2a-x)\,=-f(x) \\ \end{matrix} \right.}\] \[\therefore \] Integral 1st and IIIrd becomes zero. \[\therefore \] The given integral is depend upon \[{{a}_{1}}\]and \[{{a}_{3}}\]
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