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Question
\[\int\limits_0^{2\pi} \cos^7 x dx\]
Solution
\[Let, I = \int_0^{2\pi} \cos^7 x d x ..............(1)\]
\[ = \int_0^{2\pi} \cos^7 \left( 2\pi - x \right) d x\]
\[ = \int_0^{2\pi} - \cos^7 x d x\]
\[ \Rightarrow I = - \int_0^{2\pi} \cos^7 x d x ..............(2)\]
Adding (1) and (2) we get,
\[ 2I = \int_0^{2\pi} \cos^7 x d x - \int_0^{2\pi} \cos^7 x d x\]
\[ \Rightarrow 2I = 0\]
\[ \therefore I = 0\]
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