https://file.aiquickdraw.com/mathbot/file/1727041903130ol67xdtg.jpg

The integral provided in the image is:

\[
\int \frac{3x^2 - 2x + 5}{(x - 2)^2 (3x - 1)} \, dx
\]

To solve this integral, we can attempt a decomposition using **partial fraction decomposition**. Here’s a breakdown of how to approach it:

### Step 1: Partial Fraction Decomposition
We want to decompose the rational function:

\[
\frac{3x^2 - 2x + 5}{(x - 2)^2 (3x - 1)}
\]

into simpler fractions. A possible form of the decomposition would be:

\[
\frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{3x - 1}
\]

### Step 2: Set up the equation
Multiply both sides by \((x - 2)^2(3x - 1)\) to eliminate the denominators:

\[
3x^2 - 2x + 5 = A(x - 2)(3x - 1) + B(3x - 1) + C(x - 2)^2
\]

Now expand both sides and compare coefficients of powers of \(x\) to solve for \(A\), \(B\), and \(C\).

Would you like me to continue with the decomposition process and solve for \(A\), \(B\), and \(C\)? 

Here are some follow-up questions you might be interested in:
1. How does partial fraction decomposition work?
2. Can this integral be solved using any other method?
3. How do you compare coefficients in partial fraction decomposition?
4. Why is partial fraction decomposition useful for solving rational integrals?
5. Are there alternative ways to solve integrals involving quadratic denominators?

**Tip:** For complex fractions, partial fraction decomposition can significantly simplify the integration process.

∫ (3x^2 - 2x + 5) / ((x - 2)^2 * (3x - 1)) dx

Learn how to solve the integral of (3x^2 - 2x + 5) / ((x - 2)^2 (3x - 1)) using partial fraction decomposition. This problem covers advanced calculus techniques and integration of rational functions.

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