\[\large F(x) = \int^{x^3}_{x^2} \ln t \ dt \]

\(F(x)\) is defined as above for \(x>0\). If \(F'(2) = a \ln b\), where \(a\) and \(b\) are positive integers with \(b\) being a prime, find \(a+b\).

**Note:** \(F'(x)\) denotes the first derivative of \(F(x)\) with respect to \(x\).

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