Quadratic meets calculus!

Calculus Level pending

If \(a\) be a positive integer, the number of values of \(a\) satisfying

\(\quad \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left[ { a }^{ 2 }\left( \frac { cos(3x) }{ 4 } +\frac { 3 }{ 4 } cos(x) \right) +a\cdot sin(x)-20\cdot cos(x) \right] } dx\le -\frac { { a }^{ 2 } }{ 3 } \) is:

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