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

Options

A. `log(2/3)`
B. `log(1/4)`
C. `log(4/3)`
D. `log(8/3)`

Answer

To solve the integral \( \int_{2}^{3} \frac{dx}{x^2 - x} \), 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:
\[ \int_{2}^{3} \frac{dx}{x(x - 1)} \]

### Step 2: Partial Fraction Decomposition
Next, we will use partial fraction decomposition to express \( \frac{1}{x(x - 1)} \) in a simpler form:
\[ \frac{1}{x(x - 1)} = \frac{A}{x} + \frac{B}{x - 1} \]
Multiplying through by the denominator \( x(x - 1) \) gives:
\[ 1 = A(x - 1) + Bx \]
Expanding this, we have:
\[ 1 = Ax - A + Bx \]
Combining like terms:
\[ 1 = (A + B)x - A \]
Setting up the system of equations:
1. \( A + B = 0 \)
2. \( -A = 1 \)

From the second equation, we find \( A = -1 \). Substituting \( A \) into the first equation gives \( -1 + B = 0 \), thus \( B = 1 \).

So, we can rewrite the integral:
\[ \frac{1}{x(x - 1)} = \frac{-1}{x} + \frac{1}{x - 1} \]

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

### Step 4: Evaluate the Integrals
Now we evaluate each integral separately.

1. For \( \int \frac{-1}{x} \, dx \):
\[ \int \frac{-1}{x} \, dx = -\ln |x| \]
Evaluating from 2 to 3:
\[ -\ln |3| + \ln |2| = \ln \left( \frac{2}{3} \right) \]

2. For \( \int \frac{1}{x - 1} \, dx \):
\[ \int \frac{1}{x - 1} \, dx = \ln |x - 1| \]
Evaluating from 2 to 3:
\[ \ln |3 - 1| - \ln |2 - 1| = \ln 2 - \ln 1 = \ln 2 \]

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

### Final Answer
Thus, the value of the integral is:
\[ \int_{2}^{3} \frac{dx}{x^2 - x} = \ln \left( \frac{4}{3} \right) \]

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