Prove that \( \mathrm{P}^{\prime 9} \) \[ \int_{0}^{\frac{\pi}{2}} ...

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Prove that
\( \mathrm{P}^{\prime 9} \)
\[
\int_{0}^{\frac{\pi}{2}} \frac{d x}{\left(a^{2} \sin ^{2} x+b^{2} \cos ^{2} x\right)^{2}}=\frac{\pi\left(a^{2}+b^{2}\right)}{4 a^{3} b^{3}}
\]
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