Will you integrate for each element?

Calculus Level 4

\[\large a_k = \displaystyle \int_{0}^{\frac{\pi}{2}} (k + \sin\theta)^2\cos\theta \, d\theta\]

For \(a_k\), where \(k\) is a positive integer, as defined above, then the determinant

\[\begin{vmatrix} a_k & k & k^2 \\ a_{k+1} & k+1 & (k+1)^2 \\ a_{k+2} & k+2 & (k+2)^2 \end{vmatrix} = \frac pq \]

where \(p\) and \(q\) are coprime positive integers. Find \(p + q\).


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