# 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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