The problem asks which integral resembles $\int x^2 \sqrt{x^2 - 9} \, dx$. We need to identify the correct substitution and corresponding integral form from the given options.

1. Problem Description

The problem asks which integral resembles $\int x^2 \sqrt{x^2 - 9} \, dx$. We need to identify the correct substitution and corresponding integral form from the given options.

2. Solution Steps

We can rewrite the integral as $\int x^2 \sqrt{x^2 - 3^2} \, dx$. This suggests a substitution of the form $u = x$ and $a = 3$. Thus, we can let $a^2 = 9$. The integral then becomes $\int u^2 \sqrt{u^2 - a^2} \, du$. Now we need to compare this to the given options.

Option A: $\int u^2 \sqrt{a^2 + u^2} \, du$. This does not match the form.

Option B: $\int \sqrt{a^2 + u^2} \, du$. This does not match the form.

Option C: $\int \sqrt{u^2 - a^2} \, du$. This does not match the form.

Option D: $\int u^2 \sqrt{u^2 - a^2} \, du$. This matches the form we derived from the original integral.

Now let's check if the solution for option D is correct. According to integral tables or online calculators, the integral of the form $\int u^2 \sqrt{u^2 - a^2} du$ has the solution:

$\int u^2 \sqrt{u^2 - a^2} \, du = \frac{u}{8} (2u^2 - a^2) \sqrt{u^2 - a^2} - \frac{a^4}{8} \ln |u + \sqrt{u^2 - a^2}| + C$

This matches option D.

3. Final Answer

D. $u^2\sqrt{u^2 - a^2}du = \frac{u}{8} (2u^2 - a^2) \sqrt{u^2 - a^2} - \frac{a^4}{8} \ln |u + \sqrt{u^2 - a^2}| + C$