If `int (2cos x - sinx + lambda)/(cosx + sin x - 2) dx = A lnabs(cosx + sinx - 2) + Bx + C`. Then, ordered triplet `(A,B,lambda)` is ,

Options

A. `(1/2, 3/2, -1)`
B. `(3/2,1/2,-1)`
C. `(1/2,-1,-3/2)`
D. None of these

Answer

To solve the given problem, we need to find the ordered triplet \((A, B, \lambda)\) such that:

\[ \int \frac{2 \cos x - \sin x + \lambda}{\cos x + \sin x - 2} \, dx = A \ln |\cos x + \sin x - 2| + Bx + C \]

### Step 1: Differentiate Both Sides

We start by differentiating both sides of the equation. The left-hand side becomes:

\[ \frac{2 \cos x - \sin x + \lambda}{\cos x + \sin x - 2} \]

The right-hand side, after differentiating, becomes:

\[ \frac{A}{\cos x + \sin x - 2} \cdot (\cos x + \sin x - 2)' + B \]

Calculating the derivative of \(\cos x + \sin x - 2\):

\[ (\cos x + \sin x - 2)' = -\sin x + \cos x \]

Thus, the right-hand side becomes:

\[ \frac{A(-\sin x + \cos x)}{\cos x + \sin x - 2} + B \]

### Step 2: Set the Two Sides Equal

Now we set the two sides equal:

\[ \frac{2 \cos x - \sin x + \lambda}{\cos x + \sin x - 2} = \frac{A(-\sin x + \cos x)}{\cos x + \sin x - 2} + B \]

### Step 3: Clear the Denominator

Multiplying both sides by \(\cos x + \sin x - 2\):

\[ 2 \cos x - \sin x + \lambda = A(-\sin x + \cos x) + B(\cos x + \sin x - 2) \]

Expanding the right-hand side:

\[ A(-\sin x + \cos x) + B(\cos x + \sin x - 2) = (A + B) \cos x + (B - A) \sin x - 2B \]

### Step 4: Equate Coefficients

Now we equate coefficients from both sides:

1. Coefficient of \(\cos x\):
\[ 2 = A + B \quad \text{(1)} \]

2. Coefficient of \(\sin x\):
\[ -1 = B - A \quad \text{(2)} \]

3. Constant term:
\[ \lambda = -2B \quad \text{(3)} \]

### Step 5: Solve the System of Equations

From equation (2), we can express \(B\) in terms of \(A\):

\[ B = A - 1 \]

Substituting \(B\) in equation (1):

\[ 2 = A + (A - 1) \implies 2 = 2A - 1 \implies 2A = 3 \implies A = \frac{3}{2} \]

Now substituting \(A\) back to find \(B\):

\[ B = \frac{3}{2} - 1 = \frac{1}{2} \]

Now substituting \(B\) into equation (3) to find \(\lambda\):

\[ \lambda = -2 \left(\frac{1}{2}\right) = -1 \]

### Final Result

Thus, we have:

\[ A = \frac{3}{2}, \quad B = \frac{1}{2}, \quad \lambda = -1 \]

The ordered triplet \((A, B, \lambda)\) is:

\[ \left(\frac{3}{2}, \frac{1}{2}, -1\right) \]