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Find the integral from -2 to 1 of x(x2 – 1)9

by RoRi

Evaluate the following integral:

    \[ \int_{-2}^1x(x^2-1)^9 \, dx. \]

We make the substitution

    \[ u = x^2 - 1 \qquad \implies \qquad du = 2x. \]

The bounds of integration are then

    \[ u(-2) = 3 \qquad \text{and} \qquad u(1) = 0. \]

So we have

    \begin{align*} \int_{-2}^1 x (x^2 - 1)^9 \, dx &= \frac{1}{2} \int_3^0 u^9 \, du \\ &= \frac{1}{2} \left( \frac{u^{10}}{10} \Big \rvert_3^0 \right) \\ &= - \frac{3^{10}}{20}. \end{align*}

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