A) \[\frac{{{\pi }^{2}}}{512}\]
B) \[\frac{3{{\pi }^{2}}}{512}\]
C) \[\frac{5{{\pi }^{2}}}{512}\]
D) \[\frac{7{{\pi }^{2}}}{512}\]
Correct Answer: B
Solution :
[b] \[I=\int\limits_{0}^{\pi }{x.{{\sin }^{6}}xco{{s}^{4}}xdx}\] \[=\int\limits_{0}^{\pi }{(\pi -x).{{\sin }^{6}}(\pi -x)co{{s}^{4}}(\pi -x)dx}\] \[=\int\limits_{0}^{\pi }{(\pi -x).{{\sin }^{6}}x.\cos x.dx}\] \[=\pi \int\limits_{0}^{\pi }{{{\sin }^{6}}x.{{\cos }^{4}}x.dx}-\int\limits_{0}^{\pi }{x.{{\sin }^{6}}x.{{\cos }^{4}}x.dx}\]\[\Rightarrow I=2\times \pi \int\limits_{0}^{\pi /2}{{{\sin }^{6}}x.{{\cos }^{4}}x.dx}-I\] \[\Rightarrow I+I=2\times \pi \int\limits_{0}^{\pi /2}{{{\sin }^{6}}x.{{\cos }^{4}}x.dx}\] \[\Rightarrow 2I=2\pi \times \frac{5\times 3\times 1\times 3\times 1}{10\times 8\times 6\times 4\times 2}\times \frac{\pi }{2}\] (By Reduction formula) \[=\frac{3{{\pi }^{2}}}{2\times 8\times 2\times 8}=\frac{3{{\pi }^{2}}}{256}\] \[\therefore I=\frac{3{{\pi }^{2}}}{2\times 256}=\frac{3{{\pi }^{2}}}{512}\] Hence, option [b] is correct.
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