We are tasked with integrating the expression:

\[
\int \frac{6x^2 + 5x - 6}{x^3 + 2x^2} \, dx
\]

### Step 1: Simplify the expression
We can start by factoring the denominator:

\[
x^3 + 2x^2 = x^2(x + 2)
\]

Thus, the integral becomes:

\[
\int \frac{6x^2 + 5x - 6}{x^2(x + 2)} \, dx
\]

Now, separate the terms in the numerator:

\[
\frac{6x^2 + 5x - 6}{x^2(x + 2)} = \frac{6x^2}{x^2(x+2)} + \frac{5x}{x^2(x+2)} - \frac{6}{x^2(x+2)}
\]

This simplifies to:

\[
\frac{6}{x+2} + \frac{5}{x(x+2)} - \frac{6}{x^2(x+2)}
\]

### Step 2: Break into separate integrals
Now we can integrate each term separately:

\[
\int \left( \frac{6}{x+2} + \frac{5}{x(x+2)} - \frac{6}{x^2(x+2)} \right) dx
\]

#### First term: \(\int \frac{6}{x+2} \, dx\)
This is a simple logarithmic integral:

\[
6 \ln|x+2|
\]

#### Second term: \(\int \frac{5}{x(x+2)} \, dx\)
We can use partial fraction decomposition here. Set:

\[
\frac{5}{x(x+2)} = \frac{A}{x} + \frac{B}{x+2}
\]

Multiplying both sides by \(x(x+2)\), we get:

\[
5 = A(x+2) + Bx
\]

Expanding this gives:

\[
5 = Ax + 2A + Bx
\]

Now, combine the terms involving \(x\):

\[
5 = (A + B)x + 2A
\]

From this, we can set up the system of equations:

1. \(A + B = 0\)
2. \(2A = 5\)

Solving these gives \(A = \frac{5}{2}\) and \(B = -\frac{5}{2}\). Thus:

\[
\frac{5}{x(x+2)} = \frac{5/2}{x} - \frac{5/2}{x+2}
\]

So the integral becomes:

\[
\int \frac{5}{x(x+2)} \, dx = \frac{5}{2} \ln|x| - \frac{5}{2} \ln|x+2|
\]

#### Third term: \(\int \frac{-6}{x^2(x+2)} \, dx\)
We can use partial fraction decomposition again. Set:

\[
\frac{-6}{x^2(x+2)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+2}
\]

Multiplying both sides by \(x^2(x+2)\), we get:

\[
-6 = A x(x+2) + B(x+2) + Cx^2
\]

Expanding the right-hand side:

\[
-6 = A(x^2 + 2x) + B(x+2) + Cx^2
\]

Now, group the terms involving powers of \(x\):

\[
-6 = (A + C)x^2 + (2A + B)x + 2B
\]

From this, we can set up the system of equations:

1. \(A + C = 0\)
2. \(2A + B = 0\)
3. \(2B = -6\)

Solving these equations gives \(B = -3\), \(A = \frac{3}{2}\), and \(C = -\frac{3}{2}\). Thus:

\[
\frac{-6}{x^2(x+2)} = \frac{3/2}{x} - \frac{3}{x^2} - \frac{3/2}{x+2}
\]

Now, the integral becomes:

\[
\int \frac{-6}{x^2(x+2)} \, dx = \frac{3}{2} \ln|x| + \frac{3}{x} - \frac{3}{2} \ln|x+2|
\]

### Step 3: Combine all results
Now, we combine the results of all three integrals:

\[
\int \frac{6x^2 + 5x - 6}{x^3 + 2x^2} \, dx = 6 \ln|x+2| + \frac{5}{2} \ln|x| - \frac{5}{2} \ln|x+2| + \frac{3}{2} \ln|x| + \frac{3}{x} - \frac{3}{2} \ln|x+2|
\]

Combining the logarithmic terms:

\[
= \left(\frac{8}{2} \ln|x+2| - \frac{8}{2} \ln|x+2|\right) + \left(\frac{8}{2} \ln|x|\right) + \frac{3}{x}
\]

So the final answer is:

\[
\boxed{\frac{4}{2} \ln|x| + \frac{3}{x} + C}


Learn how to integrate (6x^2 + 5x - 6)/(x^3 + 2x^2) step-by-step using partial fraction decomposition and logarithmic integrals. This problem involves key calculus concepts, including integration techniques and partial fraction methods. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation