We are asked to evaluate the integral:

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

This is a standard integral, and it can be solved using the following formula:

$\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C$

where $a$ is a constant and $C$ is the constant of integration.

Thus, the result of the given integral is:

$\frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C$

Would you like further clarification or more details on any steps?

Here are 5 related questions to explore further:

1. How can this integral be applied in real-world problems?
2. What is the geometric interpretation of the arctangent function in this context?
3. How does the result change if $a$ is a function of $x$?
4. Can we apply substitution techniques to solve similar integrals?
5. What are the differences between this and the integral $\int \frac{1}{a^2 - x^2} \, dx$?

Tip: For integrals involving quadratic terms like $a^2 + x^2$, trigonometric substitution often simplifies the problem.