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The given expression is:

\[
\int \frac{2x^2 + 7x - 3}{x - 2} \, dx
\]

### Solution Approach:

We will solve this by performing polynomial long division on \( \frac{2x^2 + 7x - 3}{x - 2} \), since the degree of the numerator is higher than that of the denominator.

#### Step 1: Perform polynomial long division

We divide \( 2x^2 + 7x - 3 \) by \( x - 2 \).

- Divide the leading terms: \( \frac{2x^2}{x} = 2x \).
- Multiply \( 2x \) by \( x - 2 \), which gives \( 2x^2 - 4x \).
- Subtract \( (2x^2 - 4x) \) from \( (2x^2 + 7x - 3) \) to get:
  
  \[
  (2x^2 + 7x - 3) - (2x^2 - 4x) = 11x - 3
  \]

- Now, divide \( 11x \) by \( x \), which gives \( 11 \).
- Multiply \( 11 \) by \( x - 2 \), which gives \( 11x - 22 \).
- Subtract \( (11x - 22) \) from \( (11x - 3) \) to get:
  
  \[
  (11x - 3) - (11x - 22) = 19
  \]

So, the result of the division is:

\[
2x + 11 + \frac{19}{x - 2}
\]

#### Step 2: Integrate the result

Now, the integral becomes:

\[
\int \left(2x + 11 + \frac{19}{x - 2}\right) dx
\]

We can integrate this term by term:

- \( \int 2x \, dx = x^2 \)
- \( \int 11 \, dx = 11x \)
- \( \int \frac{19}{x - 2} \, dx = 19 \ln|x - 2| \)

Thus, the solution to the integral is:

\[
x^2 + 11x + 19 \ln|x - 2| + C
\]

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

### Final Answer:

\[
\boxed{x^2 + 11x + 19 \ln|x - 2| + C}
\]

Let me know if you need further clarification or additional details!

Here are 5 related questions for further exploration:
1. How would the solution change if the denominator was \( x + 2 \) instead of \( x - 2 \)?
2. How would you integrate \( \frac{x^2 + 7x - 3}{x - 1} \)?
3. What happens if the numerator and denominator are both cubic polynomials?
4. How would you integrate a rational function where the numerator is of a lower degree than the denominator?
5. How does the natural logarithm \( \ln|x - 2| \) appear in this integration?

**Tip:** Polynomial long division is essential when the degree of the numerator exceeds the degree of the denominator in a rational function.

Step-by-step solution for integrating the rational function (2x^2 + 7x - 3) / (x - 2). Learn how to apply polynomial long division followed by integration, including the natural logarithm component. Suitable for advanced high school students in Grades 11-12.

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