`int(dx)/(sin x(3+2cosx))`

Explanation

To solve the integral

\[ I = \int \frac{dx}{\sin x (3 + 2 \cos x)}, \]

we will use the method of partial fractions. Here’s a step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:

\[ I = \int \frac{dx}{\sin x (3 + 2 \cos x)}. \]

### Step 2: Rationalize the Denominator
To simplify the integral, we can multiply the numerator and denominator by \(\sin x\):

\[ I = \int \frac{\sin x \, dx}{\sin^2 x (3 + 2 \cos x)}. \]

### Step 3: Substitute for \(\cos x\)
Let \(T = \cos x\). Then, the derivative is:

\[ dT = -\sin x \, dx \quad \Rightarrow \quad \sin x \, dx = -dT. \]

### Step 4: Rewrite \(\sin^2 x\)
Using the identity \(\sin^2 x = 1 - \cos^2 x\), we can express \(\sin^2 x\) in terms of \(T\):

\[ \sin^2 x = 1 - T^2. \]

### Step 5: Substitute in the Integral
Now substituting everything into the integral, we have:

\[ I = \int \frac{-dT}{(1 - T^2)(3 + 2T)}. \]

### Step 6: Partial Fraction Decomposition
We can express the integrand using partial fractions:

\[ \frac{1}{(1 - T^2)(3 + 2T)} = \frac{A}{1 - T} + \frac{B}{1 + T} + \frac{C}{3 + 2T}. \]

### Step 7: Solve for Coefficients
Multiply through by the denominator \((1 - T^2)(3 + 2T)\) and equate coefficients to find \(A\), \(B\), and \(C\).

### Step 8: Integrate Each Term
Once we have \(A\), \(B\), and \(C\), we can integrate each term separately:

\[ I = A \int \frac{dT}{1 - T} + B \int \frac{dT}{1 + T} + C \int \frac{dT}{3 + 2T}. \]

### Step 9: Compute the Integrals
The integrals are:

\[ \int \frac{dT}{1 - T} = -\log|1 - T|, \quad \int \frac{dT}{1 + T} = \log|1 + T|, \quad \int \frac{dT}{3 + 2T} = \frac{1}{2} \log|3 + 2T|. \]

### Step 10: Combine Results
Combine the results and substitute back \(T = \cos x\):

\[ I = A(-\log|1 - \cos x|) + B(\log|1 + \cos x|) + C\left(\frac{1}{2} \log|3 + 2 \cos x|\right) + C. \]

### Step 11: Final Answer
Thus, the final answer will be in terms of \(x\):

\[ I = \frac{1}{10} \log|1 - \cos x| - \frac{1}{2} \log|1 + \cos x| + \frac{4}{5} \left(\frac{1}{2} \log|3 + 2 \cos x|\right) + C. \]