A) \[\frac{\pi }{4}\]
B) \[\frac{1}{102}\]
C) \[{{\left( \frac{\pi }{3} \right)}^{101}}\]
D) \[0\]
E) \[-1\]
Correct Answer: D
Solution :
Let \[I=\int_{0}^{\pi }{{{\cos }^{101}}x}\,dx=\int_{0}^{\pi }{{{[\cos (\pi -x)]}^{101}}}dx\] \[=\int_{0}^{\pi }{-{{\cos }^{101}}x}dx\] \[\Rightarrow \] \[2I=\int_{0}^{\pi }{({{\cos }^{101}}x-{{\cos }^{101}}x)}\,dx\] \[\Rightarrow \] \[I=0\]
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