Which Part Should I Integrate?

Calculus Level 3

Suppose that \(f\) is a smooth function (meaning that all orders of derivatives exist and are continuous) such that \( f(0)=1\), \(f(2)=3 \) and \( f' (2) = 5 \), then evaluate

\[ \large \int_{0}^{1} x f'' (2x)\, dx . \]

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