Quadratic meets calculus!

Calculus Level pending

If a a be a positive integer, the number of values of a a satisfying

∫ 0 π 2 [ a 2 ( c o s ( 3 x ) 4 + 3 4 c o s ( x ) ) + a ⋅ s i n ( x ) − 20 ⋅ c o s ( x ) ] d x ≤ − a 2 3 \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:

four three two only one

Vivek Nigam by Vivek Nigam , 22, India

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