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 identify the values of \(A\) and \(B\).

### Step 1: Identify coefficients
We can compare the integral to the standard form:

\[ \int \frac{A\cos x + B\sin x}{C\cos x + D\sin x} \, dx \]

where \(A = 3\), \(B = 2\), \(C = 5\), and \(D = 4\).

### Step 2: Calculate \(A\)
Using the formula for \(A\):

\[ A = \frac{AC + BD}{C^2 + D^2} \]

Substituting the values:

\[ A = \frac{(3)(5) + (2)(4)}{5^2 + 4^2} = \frac{15 + 8}{25 + 16} = \frac{23}{41} \]

### Step 3: Calculate \(B\)
Using the formula for \(B\):

\[ B = \frac{AD - BC}{C^2 + D^2} \]

Substituting the values:

\[ B = \frac{(3)(4) - (2)(5)}{5^2 + 4^2} = \frac{12 - 10}{25 + 16} = \frac{2}{41} \]

### Conclusion
Thus, we have:

\[ A = \frac{23}{41}, \quad B = \frac{2}{41} \]

### Final Answer
The values of \(A\) and \(B\) are:

\[ A = \frac{23}{41}, \quad B = \frac{2}{41} \]