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Evaluate the integral of x2 / (x2 + 2x + 2)2

Compute the following integral.

    \[ \int \frac{x^2 \, dx}{(x^2+2x+2)^2}. \]


We can compute the integral as follows:

    \begin{align*}  \int \frac{x^2 \, dx}{(x^2+2x+2)^2} &= \int \frac{x^2+2x+2}{(x^2+2x+2)^2} \, dx - \int \frac{2x + 2}{(x^2+2x+2)^2} \, dx \\  &= \int \frac{dx}{x^2+2x+2} - \int \frac{1}{u^2} \, du \intertext{where $u = x^2 + 2x + 2$ and $du = 2x + 2 \, dx$.  Then,}  &= \int \frac{1}{(x+1)^2+1} \, dx + \frac{1}{u} \\  &= \arctan (x+1) + \frac{1}{x^2+2x+2} + C \\  &= \frac{1}{x^2+2x+2} + \arctan (x+1) + C. \end{align*}

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