The integral of the given function is:

$\int \frac{x^2 + 1}{x^2 - 5x + 6} \, dx = x + 10 \ln |x - 3| - 5 \ln |x - 2| + C$

where $C$ is the constant of integration.

Would you like further details or clarifications?

Here are 5 related questions for further exploration:

1. How do you find the partial fraction decomposition of a rational function?
2. What are the general steps to solve an integral involving logarithmic functions?
3. How can you verify the result of an integral?
4. What are some methods to deal with complex roots in integrals?
5. How would the integration change if the denominator were a cubic expression?

Tip: When working with integrals of rational functions, check if the denominator can be factored into simpler terms. This helps in applying methods like partial fraction decomposition.