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Compute an integral of (x^2 -3)^3

Compute

    \[ \int_{-3}^3 (x^2 -3)^3 \, dx. \]


We expand and compute using the formula for the integral of a polynomial,

    \begin{align*}  \int_{-3}^3 (x^2-3)^3 \, dx &= \int_{-3}^3 (x^6 - 9x^4 + 27x^2 -27) \, dx \\  &= \left. \left( \frac{x^7}{7} - 9 \frac{x^5}{5} + 27 \frac{x^3}{3} - 27x \right) \right|_{-3}^3 \\  &= \left( \frac{2187}{7} - \frac{2187}{5}+ 243 - 81 \right) - \left( -\frac{2187}{7} + \frac{2187}{5} - 243 + 81\right) \\  &= \frac{1296}{35} + \frac{1296}{35} \\  &= \frac{2592}{35}. \end{align*}

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