$$ \int\frac{-45x^2-27x+18}{x^3+2x^2}\differentialD x $$

To evaluate the integral 

$$ \int \frac{-45x^2 - 27x + 18}{x^3 + 2x^2} \, dx, $$

we'll break the process down into steps.

### Step 1: Factor the denominator
The denominator $ x^3 + 2x^2 $ can be factored as:

$$ x^2(x + 2). $$

Thus, the integral becomes:

$$ \int \frac{-45x^2 - 27x + 18}{x^2(x + 2)} \, dx. $$

### Step 2: Partial fraction decomposition
We aim to decompose the fraction:

$$ \frac{-45x^2 - 27x + 18}{x^2(x + 2)} $$

into simpler fractions of the form:

$$ \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 2}. $$

Multiplying both sides by the denominator $ x^2(x + 2) $, we get:

$$ -45x^2 - 27x + 18 = A x(x + 2) + B(x + 2) + Cx^2. $$

Now, expand the right-hand side:

$$ A x(x + 2) = A x^2 + 2A x, $$
$$ B(x + 2) = B x + 2B, $$
$$ C x^2 = C x^2. $$

Combining all terms:

$$ A x^2 + 2A x + B x + 2B + C x^2. $$

Group terms by powers of $x$:

$$ (A + C)x^2 + (2A + B)x + 2B. $$

### Step 3: Equate coefficients
Now, equate the coefficients of powers of $x$ from both sides:

1. For $x^2$: $ A + C = -45 $,
2. For $x^1$: $ 2A + B = -27 $,
3. For $x^0$: $ 2B = 18 $.

### Step 4: Solve the system of equations
From equation (3), solve for $B$:

$$ 2B = 18 \implies B = 9. $$

Substitute $B = 9$ into equation (2):

$$ 2A + 9 = -27 \implies 2A = -36 \implies A = -18. $$

Substitute $A = -18$ into equation (1):

$$ -18 + C = -45 \implies C = -27. $$

### Step 5: Rewrite the integral
Now we can rewrite the fraction as:

$$ \frac{-45x^2 - 27x + 18}{x^2(x + 2)} = \frac{-18}{x} + \frac{9}{x^2} + \frac{-27}{x + 2}. $$

Thus, the integral becomes:

$$ \int \left( \frac{-18}{x} + \frac{9}{x^2} + \frac{-27}{x + 2} \right) dx. $$

### Step 6: Integrate term by term
Now integrate each term separately:

1. $ \int \frac{-18}{x} \, dx = -18 \ln |x|, $
2. $ \int \frac{9}{x^2} \, dx = -\frac{9}{x}, $
3. $ \int \frac{-27}{x + 2} \, dx = -27 \ln |x + 2|. $

### Step 7: Final solution
Thus, the final solution to the integral is:

$$ -18 \ln |x| - \frac{9}{x} - 27 \ln |x + 2| + C, $$

where $C$ is the constant of integration.

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Do you want more details or have any questions?  
Here are 5 questions you can explore next:

1. How does partial fraction decomposition work for higher-degree polynomials?
2. Can you try integrating a similar rational function without factorable denominators?
3. How do we know that partial fractions is a good method for this integral?
4. What is the process for factoring higher-degree polynomials?
5. Can partial fractions be applied to improper fractions?

**Tip:** Always try factoring the denominator first in rational integrals; this often simplifies the problem greatly!

Evaluate the integral: $$ \int \frac{-45x^2 - 27x + 18}{x^3 + 2x^2} \, dx $$

Explore how to evaluate the integral of a rational function using partial fraction decomposition. The problem involves breaking down the rational function and applying the power rule and logarithmic integrals. Suitable for Grades 11-12 students learning advanced calculus techniques.

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