https://file.aiquickdraw.com/mathbot/file/1727387238663vhz12qji.jpg

It seems the image contains some text distortion when extracted. However, based on what is visible, the integral in question appears to be:

\[
I = \int \frac{x^2 - 3x + 8}{x^2 - x - 2} \, dx
\]

I will now solve this integral step-by-step.

The denominator \( x^2 - x - 2 \) can be factored as:

\[
x^2 - x - 2 = (x - 2)(x + 1)
\]

Thus, the integral becomes:

\[
I = \int \frac{x^2 - 3x + 8}{(x - 2)(x + 1)} \, dx
\]

We can solve this using partial fraction decomposition. Let's proceed with that.The integral of the given function is:

\[
I = x + 2 \ln |x - 2| - 4 \ln |x + 1| + C
\]

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

Do you need any further details on the solution or explanation of steps?

Here are some related questions you might explore:

1. How do you perform partial fraction decomposition?
2. Why do we factor the denominator when integrating rational functions?
3. What is the role of logarithms in integrals like these?
4. Can this integral be computed in another way, such as substitution?
5. What are the properties of logarithms used in the final answer?

**Tip:** Always check if the denominator of a rational function can be factored when attempting integration using partial fractions.

Determine the integral I = ∫ (x^2 - 3x + 8) / (x^2 - x - 2) dx.

Learn how to determine the integral of (x^2 - 3x + 8) / (x^2 - x - 2) using partial fraction decomposition. This solution covers step-by-step integration techniques, factorization, and the application of logarithmic properties, suitable for undergraduate calculus students.

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