To solve this problem, let's start by analyzing each part of the given expression and work on finding constants $A$ and $B$.

Given Information

The problem provides the equation: $27x^3 - 18x^2 + 12 = A(3x - 2)^2 + B(9x^2 - 12x + 8) + A(3x - 2)(9x^2 - 12x + 8)$ We are tasked to express the left-hand side (LHS) in terms of constants $A$ and $B$, and then use those constants to simplify the expression.

Steps for Solution

1. Expand the Right-Hand Side (RHS): Rewrite and expand the expression on the RHS to match the terms of $x$ in the polynomial on the LHS.

2. Equating Coefficients: After expanding, match the coefficients of corresponding powers of $x$ from both sides to form equations in terms of $A$ and $B$.

3. Solve for $A$ and $B$: Solve the resulting system of equations for $A$ and $B$.

4. Rewrite the Integral: Once we know $A$ and $B$, we substitute them back into the integral provided: $\int \frac{27x^3 - 18x^2 + 12}{(3x - 2)^2 (9x^2 - 12x + 8)} \, dx$ This expression can be simplified based on the values of $A$ and $B$.

Would you like me to continue with each of these steps in detail?

Additional Questions for Exploration

1. What are the methods for integrating rational functions?
2. How can we identify the factors needed for partial fraction decomposition?
3. Why do we equate coefficients in polynomial identities?
4. Can this method be applied to integrals with different polynomial degrees in the numerator?
5. How can substitution techniques help simplify integrals of rational expressions?

Tip

When working with partial fractions, it's essential to factor the denominator completely and use each unique factor in the decomposition to ensure accurate coefficients.