To solve the indefinite integral:
$\int \frac{1}{x^2 - 5x + 6} \, dx,$ let's break it down step by step:

Step 1: Factor the quadratic denominator

The quadratic expression $x^2 - 5x + 6$ factors as: $x^2 - 5x + 6 = (x - 2)(x - 3).$

Step 2: Set up partial fraction decomposition

We rewrite the integrand using partial fractions: $\frac{1}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3},$ where $A$ and $B$ are constants to be determined.

Step 3: Solve for $A$ and $B$

Multiply through by the denominator $(x - 2)(x - 3)$: $1 = A(x - 3) + B(x - 2).$ Expand and group terms: $1 = A x - 3A + B x - 2B = (A + B)x - (3A + 2B).$ Equating coefficients of $x$ and the constant terms gives the system: $A + B = 0, \quad -3A - 2B = 1.$ From $A + B = 0$, we have $B = -A$. Substituting $B = -A$ into $-3A - 2B = 1$: $-3A - 2(-A) = 1 \implies -3A + 2A = 1 \implies -A = 1 \implies A = -1.$ Thus, $B = -A = 1$.

Step 4: Rewrite the integrand

The integrand becomes: $\frac{1}{(x - 2)(x - 3)} = \frac{-1}{x - 2} + \frac{1}{x - 3}.$

Step 5: Integrate term by term

Now, integrate each term: $\int \frac{-1}{x - 2} \, dx + \int \frac{1}{x - 3} \, dx.$ The integrals are: $\int \frac{-1}{x - 2} \, dx = -\ln|x - 2|, \quad \int \frac{1}{x - 3} \, dx = \ln|x - 3|.$

Step 6: Combine results

The indefinite integral is: $\int \frac{1}{x^2 - 5x + 6} \, dx = -\ln|x - 2| + \ln|x - 3| + C,$ where $C$ is the constant of integration.

Final Answer:

$\boxed{\ln\left|\frac{x - 3}{x - 2}\right| + C.}$

Would you like further clarifications on any of the steps?

Here are 5 related questions to deepen understanding:

1. How does partial fraction decomposition work for higher-degree polynomials?
2. Why do we use absolute values in logarithmic integrals?
3. What happens if the denominator is irreducible (cannot be factored)?
4. Can partial fraction decomposition be applied to improper fractions?
5. How do definite integrals involving logarithms differ from indefinite integrals?

Tip: Always check if a quadratic denominator can be factored before attempting more complex methods of integration.