The given integral is \(\int_{-1}^{1}x|x|dx\).

We can split the integral into two parts based on the definition of \(|x|\):

\(|x| = \begin{cases} -x, & \text{if } x < 0 \\ x, & \text{if } x \geq 0 \end{cases}\)

So, the integral becomes:

\(\int_{-1}^{1}x|x|dx = \int_{-1}^{0}x(-x)dx + \int_{0}^{1}x(x)dx\)

\(= \int_{-1}^{0}-x^2 dx + \int_{0}^{1}x^2 dx\)

\(= \left[-\frac{x^3}{3}\right]_{-1}^{0} + \left[\frac{x^3}{3}\right]_{0}^{1}\)

\(= \left(-\frac{0^3}{3} - \left(-\frac{(-1)^3}{3}\right)\right) + \left(\frac{1^3}{3} - \frac{0^3}{3}\right)\)

\(= \left(0 - \frac{1}{3}\right) + \left(\frac{1}{3} - 0\right)\)

\(= -\frac{1}{3} + \frac{1}{3}\)

\(= 0\)