The value of $\int_{-1}^1\left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right) \mathrm{d} x$ is

Let $I = \int_{-1}^1 \left(\sqrt{1+x+x^2} - \sqrt{1-x+x^2}\right) dx$.

The integrand $f(x) = \sqrt{1+x+x^2} - \sqrt{1-x+x^2}$ satisfies $f(-x) = -f(x)$, making it an odd function.

Since the integration interval $[-1, 1]$ is symmetric about the origin, the integral of any odd function over such an interval is zero.

Therefore, $I = 0$.