If \( a \) be a positive integer, the number of values of \( a \) satisfying \( \int_{0}^{\pi / ...

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If \( a \) be a positive integer, the number of values of \( a \) satisfying
\( \int_{0}^{\pi / 2}\left\{a^{2}\left(\frac{\cos 3 x}{4}+\frac{3}{4} \cos x\right)+a \sin x-20 \cos x\right\} d x \leq-\frac{a^{2}}{3} \), is
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