Numbers!

Calculus Level 2

The numbers \(p, q, r\) such that the quadratic function \(f(x) = px^2 + qx + r\) satisfies the condition \(f'(1) = 4, f(2) + f''(2) = 13\) and \(\large \int_{0}^{1} f(x)\, dx = \dfrac{5}{3}\).

Where \(p, q, r\) are integers. Find \(p+q+r\).

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