Step 1: Analyze the integrand

Let $f(x) = \frac{x^3}{x^2 + 2|x| + 1}$. We observe the symmetry of the function over the interval $[-1, 1]$.

Step 2: Check for Odd or Even function

Recall that $f(x)$ is an odd function if $f(-x) = -f(x)$. Let us evaluate $f(-x)$:

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

Step 3: Apply the property of definite integrals

For an odd function $f(x)$, the property of definite integrals states that:

Applying this to our integral where $a = 1$: