The integral given is:

$\int \frac{x^2 + 1}{x^2 - 5x + 6} \, dx$

To solve this, we'll first try to simplify the integrand by partial fraction decomposition or by a suitable substitution if applicable.

Step 1: Factor the denominator

The denominator $x^2 - 5x + 6$ can be factored as:

$x^2 - 5x + 6 = (x - 2)(x - 3)$

Step 2: Set up partial fraction decomposition

We can express the integrand as a sum of partial fractions:

$\frac{x^2 + 1}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3} + \frac{Cx + D}{(x - 2)(x - 3)}$

However, since the degree of the numerator is equal to the degree of the denominator, we don't need the term $\frac{Cx + D}{(x - 2)(x - 3)}$. Instead, we write:

$\frac{x^2 + 1}{(x - 2)(x - 3)} = 1 + \frac{(5x - 5)}{(x - 2)(x - 3)}$

This simplifies to:

$\frac{x^2 + 1}{x^2 - 5x + 6} = 1 + \frac{5x - 5}{(x - 2)(x - 3)}$

Step 3: Integrate each term

Now, integrate term by term:

1. Integrate $1$: $\int 1 \, dx = x$

2. Integrate the fraction: $\frac{5x - 5}{(x - 2)(x - 3)} = \frac{5(x - 1)}{(x - 2)(x - 3)}$

   For this, use partial fractions: $\frac{5(x - 1)}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3}$ Solve for $A$ and $B$: $5(x - 1) = A(x - 3) + B(x - 2)$ This gives: $A = 10, \quad B = -5$

   So: $\int \frac{10}{x - 2} - \frac{5}{x - 3} \, dx = 10 \ln|x - 2| - 5 \ln|x - 3|$

Step 4: Combine all the results

Putting everything together:

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

Where $C$ is the constant of integration.

Would you like any details on a specific part of the solution, or do you have any questions?

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1. How does partial fraction decomposition work?
2. Can every rational function be decomposed into partial fractions?
3. What happens if the degree of the numerator is higher than the degree of the denominator?
4. How do logarithmic integration terms arise in integrals?
5. Can substitution be an alternative method for this integral?

Tip: Always check if the degree of the numerator is less than the degree of the denominator before applying partial fractions.