To solve the integral

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

we can proceed by simplifying the integrand and then integrating term by term. Here's a step-by-step solution:

Step 1: Factor the denominator

First, factor the quadratic expression in the denominator:

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

So, the integral becomes:

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

Step 2: Perform polynomial division

Divide $x^2 + 1$ by $x^2 - 5x + 6$ (or equivalently by $(x - 2)(x - 3)$) to simplify the integrand.

$\frac{x^2 + 1}{x^2 - 5x + 6} = 1 + \frac{5x - 5}{x^2 - 5x + 6}.$

Step 3: Decompose the fraction

Now, decompose the remaining fraction into partial fractions:

$\frac{5x - 5}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3}.$

To find $A$ and $B$, multiply both sides by $(x - 2)(x - 3)$:

$5x - 5 = A(x - 3) + B(x - 2).$

Expand and collect like terms:

$5x - 5 = (A + B)x - (3A + 2B).$

This gives the system of equations:

$A + B = 5,$ $-3A - 2B = -5.$

Solving this system:

1. From $A + B = 5$, $B = 5 - A$.
2. Substitute into $-3A - 2B = -5$:

$-3A - 2(5 - A) = -5 \implies -3A - 10 + 2A = -5 \implies -A = 5 \implies A = -5.$

Substitute $A = -5$ back into $B = 5 - A$:

$B = 5 - (-5) = 10.$

Thus,

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

Step 4: Integrate each term

Now the integral is:

$\int \left[1 + \frac{-5}{x - 2} + \frac{10}{x - 3}\right] dx.$

Integrate term by term:

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

Final answer:

Combining these, the solution is:

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

where $C$ is the constant of integration.

Would you like more details on any part of this process, or do you have any other questions?

Here are 5 related questions:

1. How do you determine the factors of a quadratic expression?
2. What is the process for performing polynomial division?
3. How can you solve systems of linear equations in two variables?
4. Why is partial fraction decomposition useful in integration?
5. What is the significance of the absolute value in logarithmic integration?

Tip: When performing partial fraction decomposition, it's essential to ensure that the degree of the numerator is less than the degree of the denominator for a proper fraction. If not, first perform polynomial division.