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Question
\[\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx\]
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Solution
\[\int_{- \pi}^\pi x^{10} \sin^7 x d x\]
\[Let f\left( x \right) = x^{10} \sin^7 x\]
\[\text{Consider }f\left( - x \right) = \left( - x \right)^{10} \sin^7 \left( - x \right) = - x^{10} \sin^7 x = - f\left( x \right)\]
Hence f(x) is an odd function
Therefore
\[ \int_{- \pi}^\pi x^{10} \sin^7 x d x = 0\]
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