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Compute an integral of (8t^3 + 6t^2 – 2t + 5)

Compute:

    \[ \int_{0}^{1/2} (8t^3 + 6t^2 - 2t + 5) \, dt. \]


Using the formula for the integral of polynomials, we compute:

    \[ \int_{0}^{1/2} (8t^3 + 6t^2 - 2t + 5) \, dt = \left. \left( 8 \frac{t^4}{4} + 6 \frac{t^3}{3} - 2 \frac{t^2}{2} + 5t\right) \right|_0^{1/2} = \left( \frac{1}{8} + \frac{1}{4} - \frac{1}{4} + \frac{5}{2}\right) - (0) = \frac{21}{8}. \]

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