Piecewise Integration with Reverse Chain Rule
Consider a piecewise function defined as follows: \[f(x) = \begin{cases} 3x+1 & \text{for } x \in [0,2] \\ -2x+9 & \text{for } x \in (2,5] \end{cases}\] Determine the indefinite integral of \(f(x)\) by integrating each segment separately and then combining the results into a single piecewise antiderivative. Ensure the antiderivative is continuous at \(x=2\).