`int_2^3 (dx)/(x^2-1)`

Answer

To solve the integral \( I = \int_2^3 \frac{dx}{x^2 - 1} \), we can follow these steps:

### Step 1: Rewrite the Denominator
The expression \( x^2 - 1 \) can be factored as \( (x - 1)(x + 1) \). Thus, we have:
\[ I = \int_2^3 \frac{dx}{(x - 1)(x + 1)} \]

### Step 2: Use Partial Fraction Decomposition
We can express \( \frac{1}{(x - 1)(x + 1)} \) using partial fractions:
\[ \frac{1}{(x - 1)(x + 1)} = \frac{A}{x - 1} + \frac{B}{x + 1} \]
Multiplying through by the denominator \( (x - 1)(x + 1) \) gives:
\[ 1 = A(x + 1) + B(x - 1) \]
Expanding this, we have:
\[ 1 = Ax + A + Bx - B = (A + B)x + (A - B) \]
Setting coefficients equal, we get the system:
1. \( A + B = 0 \)
2. \( A - B = 1 \)

### Step 3: Solve for A and B
From the first equation \( A + B = 0 \), we can express \( B = -A \). Substituting into the second equation:
\[ A - (-A) = 1 \implies 2A = 1 \implies A = \frac{1}{2} \]
Then substituting back to find \( B \):
\[ B = -\frac{1}{2} \]

### Step 4: Rewrite the Integral
Now we can rewrite the integral:
\[ I = \int_2^3 \left( \frac{1/2}{x - 1} - \frac{1/2}{x + 1} \right) dx \]
This simplifies to:
\[ I = \frac{1}{2} \int_2^3 \frac{1}{x - 1} dx - \frac{1}{2} \int_2^3 \frac{1}{x + 1} dx \]

### Step 5: Integrate Each Term
The integrals can be computed as follows:
\[ \int \frac{1}{x - 1} dx = \log |x - 1| + C \]
\[ \int \frac{1}{x + 1} dx = \log |x + 1| + C \]
Thus:
\[ I = \frac{1}{2} \left[ \log |x - 1| \right]_2^3 - \frac{1}{2} \left[ \log |x + 1| \right]_2^3 \]

### Step 6: Evaluate the Limits
Calculating the first integral:
\[ \left[ \log |x - 1| \right]_2^3 = \log |3 - 1| - \log |2 - 1| = \log 2 - \log 1 = \log 2 \]
Calculating the second integral:
\[ \left[ \log |x + 1| \right]_2^3 = \log |3 + 1| - \log |2 + 1| = \log 4 - \log 3 \]

### Step 7: Combine the Results
Now substituting back:
\[ I = \frac{1}{2} (\log 2) - \frac{1}{2} (\log 4 - \log 3) \]
This simplifies to:
\[ I = \frac{1}{2} \log 2 - \frac{1}{2} (\log 4 - \log 3) = \frac{1}{2} \log 2 - \frac{1}{2} \log 4 + \frac{1}{2} \log 3 \]
Using properties of logarithms:
\[ I = \frac{1}{2} (\log 2 + \log 3 - \log 4) = \frac{1}{2} \log \left( \frac{2 \cdot 3}{4} \right) = \frac{1}{2} \log \left( \frac{3}{2} \right) \]

### Final Result
Thus, the final answer is:
\[ I = \frac{1}{2} \log \left( \frac{3}{2} \right) \]