The integral you've provided is:

$\int \frac{dx}{(x^2 + 16)^2}$

This is a standard integral that can be solved using the method of trigonometric substitution. Here's the general approach:

1. Substitute: Use the substitution $x = 4\tan(\theta)$, which simplifies the term $x^2 + 16$.
2. Change of variable: Compute $dx$ and $x^2 + 16$ in terms of $\theta$.
3. Simplify the integral: After substituting, the integral becomes easier to handle, typically involving a secant function, and can be solved.

Would you like me to walk you through the full solution step by step? Let me know if you'd like further details or if you have any questions.

Here are five related questions you might find interesting:

1. How do we use trigonometric substitution to solve integrals involving squares?
2. What is the formula for the integral of $\frac{1}{(x^2 + a^2)^2}$?
3. Can we solve this integral using partial fraction decomposition instead of substitution?
4. How do we use the substitution $x = a \tan(\theta)$ in integrals?
5. What is the general approach for solving integrals involving even powers of quadratics?

Tip: When solving integrals of rational functions with quadratics, consider trigonometric or hyperbolic substitutions, which often simplify the expressions.