To solve the integral of \(\frac{x - 7}{5 - x} \, dx\), we can break it down into simpler components.

### Step 1: Split the fraction
The integrand can be rewritten as:
\[
\frac{x - 7}{5 - x} = \frac{- (x - 5 + 2)}{5 - x} = -\frac{(x - 5) + 2}{5 - x} = -\left(\frac{x - 5}{5 - x} + \frac{2}{5 - x}\right)
\]
This simplifies to:
\[
-\left(-1 + \frac{2}{5 - x}\right) = 1 - \frac{2}{5 - x}
\]

So, the integral becomes:
\[
\int \left(1 - \frac{2}{5 - x}\right) dx
\]

### Step 2: Integrate each term separately
1. The integral of 1 with respect to \(x\) is simply:
   \[
   \int 1 \, dx = x
   \]

2. The integral of \(\frac{2}{5 - x}\) requires a substitution:
   \[
   u = 5 - x \quad \text{so} \quad du = -dx
   \]
   This gives:
   \[
   \int \frac{2}{5 - x} dx = -2 \int \frac{1}{u} du = -2 \ln|u| + C = -2 \ln|5 - x| + C
   \]

### Step 3: Combine the results
Putting it all together, we have:
\[
\int \frac{x - 7}{5 - x} \, dx = x + 2 \ln|5 - x| + C
\]

### Final Answer:
\[
\boxed{x + 2 \ln|5 - x| + C}
\]

Would you like more details or have any questions?

Here are five related questions:
1. How do you integrate more complex rational functions?
2. What is the method of partial fractions, and when is it used?
3. How do you determine the appropriate substitution for an integral?
4. Can this integral be solved using integration by parts?
5. What are some common integrals involving logarithmic functions?

**Tip:** When faced with a rational function, always check if simple algebraic manipulation or substitution can simplify the integral.

Explore the step-by-step solution to integrate (x - 7)/(5 - x) using algebraic manipulation and substitution methods. This problem involves rational functions and logarithmic integration, suitable for advanced high school and college students.

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