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.

Learn how to solve the integral ∫ (x^2)/(1 + 16x^2) dx with a step-by-step solution using the substitution method. This tutorial covers integral calculus concepts and substitution techniques, suitable for college-level students.

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