Evaluate the following integrals: \[ \cdot \int_{0}^{\pi}\left(\frac{x}{1+\sin ^{2} x}+\cos ^{7}...

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Evaluate the following integrals:
\[
\cdot \int_{0}^{\pi}\left(\frac{x}{1+\sin ^{2} x}+\cos ^{7} x\right) d x
\]
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