A calculus problem by Rudraksh Shukla

Calculus Level 5

\[\large \int^{\pi /2}_0 (\sin{x}+a\cos{x})^3\,dx - \dfrac{4a}{\pi-2} \int^{\pi /2 }_0 x\cos{x} \,dx = 2 \]

If \( a_1, a_2\) and \( a_3 \) are the three values of \(a\) which satisfy the equation above, then find the value of \( 1000 \left(a^2_1+a^2_2+a^2_3\right) \).