We are tasked with evaluating the integral:

$\int \frac{x^2}{1 + 16x^2} \, dx$

Step 1: Simplify the Integral

We can try to simplify this expression. One approach is to separate the integral into a more manageable form by rewriting the numerator $x^2$ and the denominator $1 + 16x^2$ in a way that makes integration simpler.

Step 2: Substitution

A good substitution here is to let:

$u = 1 + 16x^2$

Thus, differentiating $u$ with respect to $x$:

$du = 32x \, dx$

or equivalently:

$\frac{du}{32} = x \, dx$

Step 3: Rewrite the Integral

Now, we can rewrite the integral in terms of $u$. First, observe that:

$x^2 = \frac{u - 1}{16}$

Substitute this and $dx = \frac{du}{32x}$ into the original integral:

$\int \frac{x^2}{1 + 16x^2} \, dx = \int \frac{\frac{u - 1}{16}}{u} \cdot \frac{du}{32x}$

This is where the substitution leads to some simplifications.