`int (dx)/((x+1)(x-2))=A log (x+1)+B log (x-2)+C`, where

Options

A. `A+B=0`
B. `AB=0`
C. `A//B=-1`
D. None of these

Answer

To solve the integral \( \int \frac{dx}{(x+1)(x-2)} \) and express it in the form \( A \log(x+1) + B \log(x-2) + C \), we will follow these steps:

### Step 1: Decompose the integrand using partial fractions
We start by expressing the integrand as:
\[ \frac{1}{(x+1)(x-2)} = \frac{A}{x+1} + \frac{B}{x-2} \]
Multiplying through by the denominator \((x+1)(x-2)\) gives:
\[ 1 = A(x-2) + B(x+1) \]
Expanding this, we have:
\[ 1 = Ax - 2A + Bx + B = (A + B)x + (-2A + B) \]
By equating coefficients, we get the system of equations:
1. \( A + B = 0 \)
2. \( -2A + B = 1 \)

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

### Step 3: Rewrite the integral
Now we can rewrite the integral:
\[ \int \frac{dx}{(x+1)(x-2)} = \int \left( \frac{-\frac{1}{3}}{x+1} + \frac{\frac{1}{3}}{x-2} \right) dx \]

### Step 4: Integrate term by term
Integrating each term:
\[ = -\frac{1}{3} \int \frac{dx}{x+1} + \frac{1}{3} \int \frac{dx}{x-2} \]
This gives us:
\[ = -\frac{1}{3} \log|x+1| + \frac{1}{3} \log|x-2| + C \]

### Step 5: Combine the logarithms
Using the properties of logarithms, we can combine the logs:
\[ = \frac{1}{3} \left( \log|x-2| - \log|x+1| \right) + C = \frac{1}{3} \log\left( \frac{|x-2|}{|x+1|} \right) + C \]

### Final Expression
Thus, we have:
\[ \int \frac{dx}{(x+1)(x-2)} = -\frac{1}{3} \log(x+1) + \frac{1}{3} \log(x-2) + C \]
Comparing this with \( A \log(x+1) + B \log(x-2) + C \), we find:
- \( A = -\frac{1}{3} \)
- \( B = \frac{1}{3} \)

### Step 6: Check the options
1. \( A + B = -\frac{1}{3} + \frac{1}{3} = 0 \) (True)
2. \( A \cdot B = -\frac{1}{3} \cdot \frac{1}{3} = -\frac{1}{9} \) (False)
3. \( \frac{A}{B} = \frac{-\frac{1}{3}}{\frac{1}{3}} = -1 \) (True)

### Conclusion
The correct options are:
- Option 1: \( A + B = 0 \) (True)
- Option 3: \( \frac{A}{B} = -1 \) (True)