The integral you need to solve is:

$\int \frac{1}{25x^2 + 1} \, dx$

This type of integral can be solved using the standard formula for the integral of the form $\int \frac{1}{a^2 x^2 + 1} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$.

Here, we can rewrite the expression in the form required by the formula. Notice that $25x^2 + 1$ is similar to $5^2x^2 + 1$, so we can apply the formula directly with $a = 5$.

Thus, the integral becomes:

$\int \frac{1}{25x^2 + 1} \, dx = \frac{1}{5} \tan^{-1}(5x) + C$

Let me know if you'd like a more detailed explanation of this solution! Would you like to proceed with more examples or questions related to integrals?

Here are some related questions:

1. What is the formula for the integral of $\frac{1}{a^2 x^2 + 1}$?
2. How do we recognize when an integral fits the inverse tangent formula?
3. Can we use substitution to solve this kind of integral?
4. What happens if the coefficient of $x^2$ in the denominator is not a perfect square?
5. What is the significance of the constant $C$ in indefinite integrals?

Tip: Always check if the denominator matches the form of a known formula for easier integration.