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- Identify integral: Step : Identify the integral to solve.We need to solve the integral of from to .
- Use integration by parts: Step : Use integration by parts, where and . and (since the integral of is ).
- Apply integration by parts formula: Step : Apply the integration by parts formula: .$= \left(\frac{\(1\)}{\(6\)}\right)x*e^{\(6\)x} - \left(\frac{\(1\)}{\(36\)}\right)e^{\(6\)x} + C.
- Evaluate definite integral: Step \(4\): Evaluate the definite integral from \(-1\) to \(2\).\(\newline\)Plug in the limits of integration:\(\newline\)\(\left[\frac{1}{6}\cdot 2\cdot e^{(6\cdot 2)} - \frac{1}{36}\cdot e^{(6\cdot 2)}\right] - \left[\frac{1}{6}\cdot (-1)\cdot e^{(6\cdot (-1))} - \frac{1}{36}\cdot e^{(6\cdot (-1))}\right]\)\(\newline\)\(= \left[\frac{1}{6}\cdot 2\cdot e^{12} - \frac{1}{36}\cdot e^{12}\right] - \left[-\frac{1}{6}\cdot e^{-6} - \frac{1}{36}\cdot e^{-6}\right]\)\(\newline\)\(= \left[\frac{12}{36}\cdot e^{12} - \frac{1}{36}\cdot e^{12}\right] + \left[\frac{6}{36}\cdot e^{-6} + \frac{1}{36}\cdot e^{-6}\right]\)\(\newline\)\(= \left[\frac{\(11\)}{\(36\)}\cdot e^{\(12\)} + \frac{\(7\)}{\(36\)}\cdot e^{\(-6\)}\right].)
