To solve the integral

$I = \int \frac{12x^3 - 8x^2 - 236x - 102}{x^2 - x - 20} \, dx$

we begin by performing polynomial long division on the numerator by the denominator.

Step 1: Polynomial Division

We are dividing $12x^3 - 8x^2 - 236x - 102$ by $x^2 - x - 20$.

1. Divide the leading terms: $\frac{12x^3}{x^2} = 12x$. Multiply $12x$ by $x^2 - x - 20$, giving: $12x(x^2 - x - 20) = 12x^3 - 12x^2 - 240x$ Subtract this from the original numerator: $(12x^3 - 8x^2 - 236x - 102) - (12x^3 - 12x^2 - 240x) = 4x^2 + 4x - 102$

2. Divide the leading terms: $\frac{4x^2}{x^2} = 4$. Multiply $4$ by $x^2 - x - 20$, giving: $4(x^2 - x - 20) = 4x^2 - 4x - 80$ Subtract this from $4x^2 + 4x - 102$: $(4x^2 + 4x - 102) - (4x^2 - 4x - 80) = 8x - 22$

Thus, the division gives: $\frac{12x^3 - 8x^2 - 236x - 102}{x^2 - x - 20} = 12x + 4 + \frac{8x - 22}{x^2 - x - 20}$

Step 2: Break Down the Integral

Now we can split the integral: $I = \int \left( 12x + 4 + \frac{8x - 22}{x^2 - x - 20} \right) \, dx$ This can be written as: $I = \int (12x + 4) \, dx + \int \frac{8x - 22}{x^2 - x - 20} \, dx$

Step 3: Solve Each Integral

1. First Integral:

$\int (12x + 4) \, dx = 6x^2 + 4x + C_1$

2. Second Integral:

We simplify $x^2 - x - 20 = (x - 5)(x + 4)$, then decompose the rational function using partial fraction decomposition: $\frac{8x - 22}{(x - 5)(x + 4)} = \frac{A}{x - 5} + \frac{B}{x + 4}$ Multiplying both sides by $(x - 5)(x + 4)$ and solving for $A$ and $B$: $8x - 22 = A(x + 4) + B(x - 5)$ Expanding and equating coefficients: $8x - 22 = A(x + 4) + B(x - 5) = (A + B)x + (4A - 5B)$ This gives two equations: $A + B = 8 \quad \text{(1)}$ $4A - 5B = -22 \quad \text{(2)}$ Solving this system of equations:

From equation (1): $A = 8 - B$. Substituting into equation (2): $4(8 - B) - 5B = -22$ $32 - 4B - 5B = -22$ $32 - 9B = -22$ $-9B = -54 \quad \Rightarrow \quad B = 6$ Substituting $B = 6$ into equation (1): $A + 6 = 8 \quad \Rightarrow \quad A = 2$

Thus, the partial fraction decomposition is: $\frac{8x - 22}{(x - 5)(x + 4)} = \frac{2}{x - 5} + \frac{6}{x + 4}$

Now, integrate each term: $\int \frac{2}{x - 5} \, dx = 2 \ln |x - 5| + C_2$ $\int \frac{6}{x + 4} \, dx = 6 \ln |x + 4| + C_3$

Step 4: Combine Results

Thus, the full solution is: $I = 6x^2 + 4x + 2 \ln |x - 5| + 6 \ln |x + 4| + C$ where $C$ is the constant of integration.

Would you like further clarification on any steps or additional details?

---

Here are 5 related questions:

1. How do you perform partial fraction decomposition for more complex denominators?
2. What are the common techniques for solving integrals with polynomial numerators and denominators?
3. How can polynomial long division be applied to integrals?
4. What is the general process for integrating rational functions?
5. How do you handle improper integrals when the degree of the numerator is higher than the denominator?

Tip: Always simplify the integrand through algebraic manipulation (such as long division or partial fractions) before attempting to integrate complex rational expressions.