If `int(2x+3)/(x^(2)-5x+6)dx`
`=9log(x-3)-7log(x-2)+A,` then A =

Options

A. `5log(x-2)`+constant
B. `-4log(x-3)+`constant
C. Constant
D. None of these

Answer

To solve the integral \( \int \frac{2x + 3}{x^2 - 5x + 6} \, dx \) and find the constant \( A \) in the equation \( \int \frac{2x + 3}{x^2 - 5x + 6} \, dx = 9 \log(x - 3) - 7 \log(x - 2) + A \), we will follow these steps:

### Step 1: Factor the Denominator
The first step is to factor the quadratic expression in the denominator:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 2: Set Up Partial Fraction Decomposition
We can express the integrand using partial fractions:
\[ \frac{2x + 3}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3} \]
where \( A \) and \( B \) are constants to be determined.

### Step 3: Solve for Constants \( A \) and \( B \)
Multiply through by the denominator \((x - 2)(x - 3)\):
\[ 2x + 3 = A(x - 3) + B(x - 2) \]
Expanding the right-hand side:
\[ 2x + 3 = Ax - 3A + Bx - 2B = (A + B)x + (-3A - 2B) \]
Now, equate coefficients:
1. For \( x \): \( A + B = 2 \)
2. For the constant term: \( -3A - 2B = 3 \)

### Step 4: Solve the System of Equations
From the first equation, we have:
\[ B = 2 - A \]
Substituting into the second equation:
\[ -3A - 2(2 - A) = 3 \]
Simplifying:
\[ -3A - 4 + 2A = 3 \implies -A - 4 = 3 \implies -A = 7 \implies A = -7 \]
Now substituting \( A = -7 \) back into \( B = 2 - A \):
\[ B = 2 - (-7) = 9 \]

### Step 5: Rewrite the Integral
Now we can rewrite the integral:
\[ \int \left( \frac{-7}{x - 2} + \frac{9}{x - 3} \right) \, dx \]

### Step 6: Integrate Each Term
Integrating each term separately:
\[ \int \frac{-7}{x - 2} \, dx + \int \frac{9}{x - 3} \, dx = -7 \log|x - 2| + 9 \log|x - 3| + C \]

### Step 7: Combine and Compare
Combining the results:
\[ = 9 \log(x - 3) - 7 \log(x - 2) + C \]
We are given that this equals \( 9 \log(x - 3) - 7 \log(x - 2) + A \). Therefore, we can equate \( C \) to \( A \).

### Step 8: Find \( A \)
Since \( C \) is an arbitrary constant of integration, we can conclude that:
\[ A = C \]
Thus, the value of \( A \) is simply the constant of integration, which we can denote as \( C \).

### Final Answer
The value of \( A \) is:
\[ A = C \]