We need to integrate the following expression:

\[
\int \left( x + \frac{4}{x^2 - 5x + 6} \right) dx
\]

### Step 1: Simplify the expression
We can begin by factoring the quadratic expression in the denominator:

\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]

Now, the integral becomes:

\[
\int \left( x + \frac{4}{(x - 2)(x - 3)} \right) dx
\]

### Step 2: Split the integral
We can now split the integral into two parts:

\[
\int x \, dx + \int \frac{4}{(x - 2)(x - 3)} \, dx
\]

### Step 3: Solve the first part
The first part is straightforward:

\[
\int x \, dx = \frac{x^2}{2}
\]

### Step 4: Solve the second part using partial fractions
To solve the second part, we decompose the fraction:

\[
\frac{4}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3}
\]

Multiplying both sides by \((x - 2)(x - 3)\), we get:

\[
4 = A(x - 3) + B(x - 2)
\]

Expanding:

\[
4 = A(x - 3) + B(x - 2) = Ax - 3A + Bx - 2B
\]

\[
4 = (A + B)x - (3A + 2B)
\]

Now, we can compare the coefficients of \(x\) and the constant terms:

1. \(A + B = 0\)
2. \(-3A - 2B = 4\)

From the first equation, \(B = -A\). Substituting this into the second equation:

\[
-3A - 2(-A) = 4 \implies -3A + 2A = 4 \implies -A = 4 \implies A = -4
\]

Therefore, \(B = 4\).

So, we can rewrite the second part of the integral as:

\[
\int \frac{4}{(x - 2)(x - 3)} \, dx = \int \left( \frac{-4}{x - 2} + \frac{4}{x - 3} \right) dx
\]

### Step 5: Solve the partial fraction integrals
Now, we integrate each term separately:

\[
\int \frac{-4}{x - 2} \, dx = -4 \ln|x - 2|
\]

\[
\int \frac{4}{x - 3} \, dx = 4 \ln|x - 3|
\]

### Step 6: Combine the results
Combining all parts, we get the final solution:

\[
\frac{x^2}{2} - 4 \ln|x - 2| + 4 \ln|x - 3| + C
\]

where \(C\) is the constant of integration.

Would you like further details or clarifications?

Here are 5 related questions to expand on this topic:
1. How do you apply partial fraction decomposition for more complex denominators?
2. What are the conditions for a function to be integrable?
3. How does substitution work in integrals involving radicals?
4. What is the geometric interpretation of definite integrals?
5. Can you solve improper integrals and how are they related to limits?

**Tip:** Always double-check your factorization when using partial fractions—it can simplify your integrals significantly!

Explore how to integrate x + 4 / (x^2 - 5x + 6) using partial fraction decomposition. This solution covers quadratic factorization, splitting integrals, and solving with partial fractions. Ideal for students in advanced high school or undergraduate calculus.

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