$\int_{-2}^2\left|x^2-x-2\right| \mathrm{dx}=$

The definite integral $\int_{-2}^{2}|x^2 - x - 2|dx$ requires analyzing the absolute value function. The quadratic $x^2 - x - 2 = (x-2)(x+1)$ has roots at $x = -1$ and $x = 2$, and is negative between these roots while positive elsewhere.

The integral splits into two parts:
$\int_{-2}^{-1}(x^2 - x - 2)dx + \int_{-1}^{2}(-x^2 + x + 2)dx$

Evaluating the first integral:
$\left[\frac{x^3}{3} - \frac{x^2}{2} - 2x\right]_{-2}^{-1} = \frac{11}{6}$

Evaluating the second integral:
$\left[-\frac{x^3}{3} + \frac{x^2}{2} + 2x\right]_{-1}^{2} = \frac{9}{2}$

The total integral is $\frac{11}{6} + \frac{9}{2} = \frac{19}{3}$.

Final answer: $\frac{19}{3}$