`int _(2) ^(3) (x)/( x ^(2) -1) dx =`

Options

A. `log ((8)/(3))`
B. `- log ((8)/(3))`
C. `1/2 log ((8)/(3))`
D. `-1/2 log ((8)/(3))`

Answer

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

### Step 1: Set up the integral
Let
\[ I = \int_{2}^{3} \frac{x}{x^2 - 1} \, dx \]

### Step 2: Substitute \( t = x^2 - 1 \)
We can use the substitution \( t = x^2 - 1 \). Then, differentiate both sides:
\[ dt = 2x \, dx \quad \Rightarrow \quad dx = \frac{dt}{2x} \]

### Step 3: Change the limits of integration
Now, we need to change the limits of integration according to our substitution:
- When \( x = 2 \):
\[ t = 2^2 - 1 = 4 - 1 = 3 \]
- When \( x = 3 \):
\[ t = 3^2 - 1 = 9 - 1 = 8 \]

### Step 4: Rewrite the integral in terms of \( t \)
Substituting \( t \) and \( dx \) into the integral, we have:
\[ I = \int_{3}^{8} \frac{x}{t} \cdot \frac{dt}{2x} = \int_{3}^{8} \frac{1}{2} \cdot \frac{dt}{t} \]
This simplifies to:
\[ I = \frac{1}{2} \int_{3}^{8} \frac{dt}{t} \]

### Step 5: Evaluate the integral
The integral \( \int \frac{dt}{t} \) is \( \log |t| \). Therefore,
\[ I = \frac{1}{2} \left[ \log |t| \right]_{3}^{8} = \frac{1}{2} \left( \log 8 - \log 3 \right) \]

### Step 6: Simplify using logarithmic properties
Using the property of logarithms \( \log a - \log b = \log \frac{a}{b} \), we get:
\[ I = \frac{1}{2} \log \frac{8}{3} \]

### Final Answer
Thus, the value of the integral is:
\[ I = \frac{1}{2} \log \frac{8}{3} \]

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