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.

Learn how to evaluate the integral ∫ 1/(a^2 + x^2) dx using the inverse trigonometric function formula. This problem focuses on key calculus concepts, including integration techniques and the application of inverse trigonometric functions. Suitable for undergraduate calculus students.

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