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Calculus Level 5

\[\left| 3\sin x - 4\cos x + t + \cfrac 4t \right| \]

For all real \(t\), let \( f(t) \) be the minimum of the above expression over \( x\in \mathbb{R} \).

Let \( \displaystyle \int_1^{10} f(t) \, dt = 4 ( a + \ln b ) \) for some rational \( a \) and \(b\).

Find the value of \( a + 2b \).

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