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$\int_{-1}^1 x|x| d x=$
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$0$
$\int_{-1}^1 x|x| d x$
$\begin{aligned} & =\int_{-1}^0 x(-x) d x+\int_0^1 x \cdot x d x \\ & =-\int_{-1}^0 x^2 d x+\int_0^1 x^2 d x \\ & =-\left[\frac{x^3}{3}\right]_{-1}^0+\left[\frac{x^3}{3}\right]_0^1 \\ & =-\left(0+\frac{1}{3}\right)+\left(\frac{1}{3}-0\right)=0\end{aligned}$
$\begin{aligned} & =\int_{-1}^0 x(-x) d x+\int_0^1 x \cdot x d x \\ & =-\int_{-1}^0 x^2 d x+\int_0^1 x^2 d x \\ & =-\left[\frac{x^3}{3}\right]_{-1}^0+\left[\frac{x^3}{3}\right]_0^1 \\ & =-\left(0+\frac{1}{3}\right)+\left(\frac{1}{3}-0\right)=0\end{aligned}$
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