Evaluate $\int_{-1}^{1} x^3 \cos x \, dx$.

Let $f(x) = x^3 \cos x$. Let's test if this function is even or odd by checking $f(-x)$:

$$f(-x) = (-x)^3 \cos(-x) = -x^3 \cos x = -f(x)$$

Since $f(-x) = -f(x)$, the integrand is an odd function.

By the definite integral property, for any odd function integrated over symmetric limits $[-a, a]$:

$$\int_{-a}^{a} f(x) \, dx = 0$$

Thus, $\int_{-1}^{1} x^3 \cos x \, dx = 0$.

Answer: 0