The integral of $\frac{1}{a^2 + x^2}$ with respect to $x$ can be solved using a standard formula for integrals of this type. The formula is:

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

where $C$ is the constant of integration.

Solution:

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

Would you like further details or have any questions on this?

Here are 5 related questions to expand on this:

1. How does the arctangent function arise in integrals like this?
2. What is the geometric interpretation of the arctangent in this context?
3. How would the integral change if we had $x^2 - a^2$ in the denominator?
4. What is the definite integral of $\frac{1}{a^2 + x^2}$ over a specific interval?
5. How can this result be applied to solving differential equations?

Tip: In problems involving $a^2 + x^2$, trigonometric substitution is a powerful tool for integration.