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Definite integral

\begin{align} \int_{ 0 }^{ \pi } \sin ^{10}\left(x\right) \,\mathrm{d}x&= \left[ {{5\,\sin \left(8\,x\right)}\over{2048}}+{{15\,\sin \left(4\,x \right)}\over{256}}-{{\sin ^5\left(2\,x\right)}\over{320}}+{{\sin ^3 \left(2\,x\right)}\over{16}}-{{\sin \left(2\,x\right)}\over{4}}+{{63 \,x}\over{256}} \right]_{ 0 }^{ \pi }\\ &= {{63\,\pi}\over{256}}-0 \\&= {{63\,\pi}\over{256}} \\& \approx 0.77312631709436 \end{align} Help to find the primitive function

Mean value

\begin{align} \mu&=\frac{1}{b-a}\int_{a}^{b}f(x)\,\mathrm{d}x\\&=\frac{ {{63\,\pi}\over{256}} }{ \pi }\\&= {{63}\over{256}} \\&\approx 0.24609375 \end{align}

Graph of the function and mean value