inegrate (3(3x+2y))/(x(x+y)) with respect to dx

We are tasked with integrating the expression:

\[
\int \frac{3(3x + 2y)}{x(x + y)} \, dx
\]

### Step 1: Simplify the expression
We start by simplifying the integrand:

\[
\frac{3(3x + 2y)}{x(x + y)} = 3 \cdot \frac{3x + 2y}{x(x + y)}
\]

Now, break up the numerator \( 3x + 2y \):

\[
3 \cdot \left( \frac{3x}{x(x + y)} + \frac{2y}{x(x + y)} \right) = 3 \left( \frac{3}{x + y} + \frac{2y}{x(x + y)} \right)
\]

### Step 2: Separate the integrals
Now, we can separate the integrals for each term:

\[
3 \int \left( \frac{3}{x + y} + \frac{2y}{x(x + y)} \right) \, dx
\]

This gives us two integrals to solve:

\[
3 \int \frac{3}{x + y} \, dx + 3 \int \frac{2y}{x(x + y)} \, dx
\]

### Step 3: Solve the first integral
The first integral is straightforward:

\[
3 \int \frac{3}{x + y} \, dx
\]

Since \( y \) is treated as a constant with respect to \( x \), let \( u = x + y \), which gives \( du = dx \). This transforms the integral into:

\[
3 \cdot 3 \int \frac{1}{u} \, du = 9 \ln |x + y|
\]

### Step 4: Solve the second integral
The second integral is:

\[
3 \int \frac{2y}{x(x + y)} \, dx
\]

Factor out constants:

\[
6y \int \frac{1}{x(x + y)} \, dx
\]

Using partial fractions decomposition:

\[
\frac{1}{x(x + y)} = \frac{A}{x} + \frac{B}{x + y}
\]

Solving for \( A \) and \( B \), we get:

\[
A = \frac{1}{y}, \quad B = -\frac{1}{y}
\]

Thus:

\[
\frac{1}{x(x + y)} = \frac{1}{y} \left( \frac{1}{x} - \frac{1}{x + y} \right)
\]

Substituting this into the integral:

\[
6y \cdot \frac{1}{y} \int \left( \frac{1}{x} - \frac{1}{x + y} \right) \, dx = 6 \left( \ln |x| - \ln |x + y| \right)
\]

### Step 5: Combine results
Now, combining both integrals, we get:

\[
9 \ln |x + y| + 6 \left( \ln |x| - \ln |x + y| \right)
\]

Simplifying this expression:

\[
3 \ln |x + y| + 6 \ln |x|
\]

### Final Answer:
\[
\int \frac{3(3x + 2y)}{x(x + y)} \, dx = 3 \ln |x + y| + 6 \ln |x| + C
\]

Would you like more details or have any questions?

Here are 5 related questions to expand this topic:
1. How do we handle integration when both variables \(x\) and \(y\) are present?
2. What is the method of partial fraction decomposition used in integration?
3. How does substitution simplify integrals with linear terms?
4. Can you explain the importance of the constant of integration?
5. How does the logarithmic function behave in integrals like this?

**Tip:** Always check if variables other than the one of integration are constants before applying simplification techniques.

Integrate (3(3x+2y))/(x(x+y)) with respect to dx

Learn how to integrate (3(3x + 2y))/(x(x + y)) with respect to x using partial fraction decomposition and substitution methods. This detailed solution involves breaking the integral into simpler parts and applying logarithmic integration. Suitable for advanced high school and early college students.

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