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Compute an integral of (x+1)^2

Compute:

    \[ \int_{-1}^0 (x+1)^2 \, dx. \]


Here, we use the fact that the integral is invariant under translation (Theorem 1.18 of Apostol),

    \[ \int_{-1}^0 (x+1)^2 \, dx  = \int_{-1+1}^{0+1} (x-1+1)^2 \, dx = \int_0^1 x^2 \, dx = \left. \frac{x^3}{3} \right|_0^1 = \frac{1}{3}. \]

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