If \( \int_{0}^{n \pi} f\left(\cos ^{2} x\right) d x=k \int_{0}^{\pi} f\left(\cos ^{2} x\right) ...

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If \( \int_{0}^{n \pi} f\left(\cos ^{2} x\right) d x=k \int_{0}^{\pi} f\left(\cos ^{2} x\right) d x \), then find the value \( k \)
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