To solve the integral of $\frac{1}{x^2 + 25}$, we can use a standard formula for integrals involving sums of squares:

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

In this case, $a = 5$, because $25 = 5^2$. So, applying the formula:

$\int \frac{1}{x^2 + 25} \, dx = \frac{1}{5} \arctan\left(\frac{x}{5}\right) + C$

Thus, the integral is:

$\frac{1}{5} \arctan\left(\frac{x}{5}\right) + C$

Would you like a detailed breakdown of this solution, or do you have any questions?

Here are five related questions to expand on this:

1. How would you approach integrating $\frac{1}{x^2 + b^2}$ for a different constant $b$?
2. What happens if the denominator was $x^2 - 25$ instead of $x^2 + 25$?
3. Can you differentiate $\frac{1}{5} \arctan\left(\frac{x}{5}\right)$ to verify the solution?
4. How does the integration change if you introduce an $x$-dependent term in the numerator?
5. What is the relationship between the arctangent function and trigonometric identities?

Tip: Always recognize sums of squares in the integrand as a cue to use inverse trigonometric functions for easier integration.