`int (2x+3)/((x+2)(x-2))dx`

Answer

To solve the integral \( \int \frac{2x + 3}{(x + 2)(x - 2)} \, dx \), we will use the method of partial fractions. Here’s the step-by-step solution:

### Step 1: Set up the partial fraction decomposition
We can express the integrand as:
\[ \frac{2x + 3}{(x + 2)(x - 2)} = \frac{A}{x + 2} + \frac{B}{x - 2} \]
where \( A \) and \( B \) are constants to be determined.

**Hint:** Start by identifying the form of the partial fraction decomposition.

### Step 2: Combine the right-hand side
Multiplying both sides by the denominator \((x + 2)(x - 2)\) gives:
\[ 2x + 3 = A(x - 2) + B(x + 2) \]

**Hint:** Ensure that you multiply through by the common denominator to eliminate the fractions.

### Step 3: Expand and rearrange
Expanding the right-hand side:
\[ 2x + 3 = Ax - 2A + Bx + 2B \]
Combining like terms:
\[ 2x + 3 = (A + B)x + (2B - 2A) \]

**Hint:** Group the terms by their degree (linear and constant).

### Step 4: Set up equations for coefficients
By comparing coefficients from both sides, we get the following system of equations:
1. \( A + B = 2 \) (coefficient of \( x \))
2. \( 2B - 2A = 3 \) (constant term)

**Hint:** Write down equations based on the coefficients of \( x \) and the constant terms.

### Step 5: Solve the system of equations
From the first equation, we can express \( B \) in terms of \( A \):
\[ B = 2 - A \]
Substituting into the second equation:
\[ 2(2 - A) - 2A = 3 \]
This simplifies to:
\[ 4 - 2A - 2A = 3 \implies 4 - 4A = 3 \implies 4A = 1 \implies A = \frac{1}{4} \]
Now substituting \( A \) back to find \( B \):
\[ B = 2 - \frac{1}{4} = \frac{8}{4} - \frac{1}{4} = \frac{7}{4} \]

**Hint:** Substitute back to find the second variable after determining the first.

### Step 6: Rewrite the integral
Now we can rewrite the integral:
\[ \int \frac{2x + 3}{(x + 2)(x - 2)} \, dx = \int \left( \frac{1/4}{x + 2} + \frac{7/4}{x - 2} \right) \, dx \]

**Hint:** Substitute the values of \( A \) and \( B \) back into the integral.

### Step 7: Integrate each term
Now we can integrate term by term:
\[ \int \frac{1/4}{x + 2} \, dx + \int \frac{7/4}{x - 2} \, dx = \frac{1}{4} \ln |x + 2| + \frac{7}{4} \ln |x - 2| + C \]

**Hint:** Remember to apply the logarithmic integration rule.

### Final Answer
Thus, the final result is:
\[ \int \frac{2x + 3}{(x + 2)(x - 2)} \, dx = \frac{1}{4} \ln |x + 2| + \frac{7}{4} \ln |x - 2| + C \]

**Hint:** Don’t forget to include the constant of integration \( C \) at the end.