If `int frac{3cosx+2sinx}{4sinx+5cosx}dx=Ax+Blog|4sinx+5cosx|+C`,then:

Options

A. `A=frac{23}{41},B=frac{1}{41}`
B. `A=frac{23}{41},B=frac{2}{41}`
C. `A=frac{11}{23},B=frac{2}{23}`
D. `A=frac{12}{23},B=frac{2}{23}`

Answer

To solve the integral \(\int \frac{3 \cos x + 2 \sin x}{4 \sin x + 5 \cos x} \, dx\) and express it in the form \(Ax + B \log |4 \sin x + 5 \cos x| + C\), we will follow these steps:

### Step 1: Identify coefficients
We compare the integral with the standard form:
\[ \int \frac{a \cos x + b \sin x}{c \cos x + d \sin x} \, dx = Ax + B \log |c \cos x + d \sin x| + C \]
From our integral, we identify:
- \(a = 3\)
- \(b = 2\)
- \(c = 5\)
- \(d = 4\)

### Step 2: Calculate \(A\)
Using the formula for \(A\):
\[ A = \frac{ad + bc}{c^2 + d^2} \]
Substituting the values:
\[ A = \frac{(3)(4) + (2)(5)}{5^2 + 4^2} = \frac{12 + 10}{25 + 16} = \frac{22}{41} \]

### Step 3: Calculate \(B\)
Using the formula for \(B\):
\[ B = \frac{ab - cd}{c^2 + d^2} \]
Substituting the values:
\[ B = \frac{(3)(2) - (5)(4)}{5^2 + 4^2} = \frac{6 - 20}{25 + 16} = \frac{-14}{41} \]

### Step 4: Conclusion
Thus, we have:
- \(A = \frac{22}{41}\)
- \(B = \frac{-14}{41}\)

The integral can be expressed as:
\[ \int \frac{3 \cos x + 2 \sin x}{4 \sin x + 5 \cos x} \, dx = \frac{22}{41}x - \frac{14}{41} \log |4 \sin x + 5 \cos x| + C \]