Integration of the defined function

Calculus Level 4

Let \(f(x)=x^3-3x-1.\) For a real number \(t\geq-1,\) define the function \[g(t)=\max_{-1\leq x\leq t}|f(x)|.\] If \(\displaystyle \int_{-1}^{1}g(t)\, dt =\frac{p}{q},\) where \(p\) and \(q\) are coprime positive integers, what is the value of \(p+q?\)

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