The integral to solve is:

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

Step-by-Step Solution:

1. Factor the denominator:
   The denominator $x^2 - 5x + 6$ can be factored as: $x^2 - 5x + 6 = (x - 2)(x - 3).$

   Thus, the integral becomes: $\int \frac{x^2 + 1}{(x - 2)(x - 3)} \, dx.$

2. Rewrite the fraction:
   Express the integrand as a sum of partial fractions: $\frac{x^2 + 1}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3} + C.$

   Multiply both sides by $(x - 2)(x - 3)$ to clear the denominators: $x^2 + 1 = A(x - 3) + B(x - 2) + C(x - 2)(x - 3).$

   To find the constants $A$, $B$, and $C$, we can substitute suitable values of $x$ or equate coefficients. Let me calculate these constants.The constants for the partial fraction decomposition are:

$A = -5, \quad B = 10, \quad C = 1.$

Thus, the expression can be rewritten as:

$\frac{x^2 + 1}{(x - 2)(x - 3)} = \frac{-5}{x - 2} + \frac{10}{x - 3} + \frac{x}{(x - 2)(x - 3)}.$

3. Integrate term by term:
   Now, the integral can be written as:

$\int \left( \frac{-5}{x - 2} + \frac{10}{x - 3} + \frac{x}{(x - 2)(x - 3)} \right) \, dx.$

Let's split it into three separate integrals:

$-5 \int \frac{1}{x - 2} \, dx + 10 \int \frac{1}{x - 3} \, dx + \int \frac{x}{(x - 2)(x - 3)} \, dx.$

The first two integrals are straightforward logarithms:

$-5 \ln|x - 2| + 10 \ln|x - 3|.$

For the third integral:

$\int \frac{x}{(x - 2)(x - 3)} \, dx,$

rewrite it in a simpler form. Since we have constants $C = 1$, this term is just the integral of $\frac{x}{(x - 2)(x - 3)}$.

To compute the remaining integral, we notice that:

$\frac{x}{(x - 2)(x - 3)} = \frac{1}{(x - 2)(x - 3)}(x).$

Let's simplify and integrate the remaining terms.The complete integral is:

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

Simplifying the expression:

$= 13 \ln|x - 3| - 7 \ln|x - 2| + C,$

where $C$ is the constant of integration.

Conclusion

The solution to the integral is:

$\int \frac{x^2 + 1}{x^2 - 5x + 6} \, dx = 13 \ln|x - 3| - 7 \ln|x - 2| + C.$

Would you like more details on any specific step, or do you have any other questions?

Related Questions:

1. How do you perform partial fraction decomposition for different types of rational functions?
2. What is the method for integrating functions involving quadratic polynomials?
3. How does the substitution method differ from partial fraction decomposition in integration?
4. Can you explain how logarithmic integration works for different kinds of expressions?
5. How do you determine the constants in partial fraction decomposition by equating coefficients?

Tip: Always factor the denominator first when attempting partial fraction decomposition to identify possible terms for decomposition!