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