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

Options

A. `log((2)/(3))`
B. `log((4)/(3))`
C. `log((8)/(3))`
D. `-log 4`

Answer

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

### Step 1: Factor the Denominator
The first step is to factor the denominator \( x^2 - x \):
\[ x^2 - x = x(x - 1) \]
Thus, we can rewrite the integral as:
\[ \int_{2}^{3} \frac{1}{x(x - 1)} \, dx \]

### Step 2: Partial Fraction Decomposition
Next, we will use partial fraction decomposition to express \( \frac{1}{x(x - 1)} \) as a sum of simpler fractions:
\[ \frac{1}{x(x - 1)} = \frac{A}{x} + \frac{B}{x - 1} \]
Multiplying through by the denominator \( x(x - 1) \) gives:
\[ 1 = A(x - 1) + Bx \]
Expanding and rearranging:
\[ 1 = Ax - A + Bx \]
\[ 1 = (A + B)x - A \]
From this, we can set up the system of equations:
1. \( A + B = 0 \)
2. \( -A = 1 \)

Solving these equations, we find:
- From \( -A = 1 \), we have \( A = -1 \).
- Substituting \( A \) into \( A + B = 0 \) gives \( -1 + B = 0 \) or \( B = 1 \).

Thus, we can rewrite the integral as:
\[ \int_{2}^{3} \left( \frac{-1}{x} + \frac{1}{x - 1} \right) \, dx \]

### Step 3: Split the Integral
Now we can split the integral into two parts:
\[ \int_{2}^{3} \left( \frac{-1}{x} \right) \, dx + \int_{2}^{3} \left( \frac{1}{x - 1} \right) \, dx \]

### Step 4: Evaluate Each Integral
Now we evaluate each integral separately.

1. For the first integral:
\[ \int \frac{-1}{x} \, dx = -\log |x| \quad \text{(evaluated from 2 to 3)} \]
\[ = -[\log 3 - \log 2] = -\log 3 + \log 2 = \log \frac{2}{3} \]

2. For the second integral:
\[ \int \frac{1}{x - 1} \, dx = \log |x - 1| \quad \text{(evaluated from 2 to 3)} \]
\[ = [\log(3 - 1) - \log(2 - 1)] = \log 2 - \log 1 = \log 2 \quad \text{(since log 1 = 0)} \]

### Step 5: Combine the Results
Now we combine the results of both integrals:
\[ \int_{2}^{3} \frac{1}{x^2 - x} \, dx = \log \frac{2}{3} + \log 2 = \log \left( \frac{2}{3} \cdot 2 \right) = \log \frac{4}{3} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{2}^{3} \frac{1}{x^2 - x} \, dx = \log \frac{4}{3} \]

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