`int(2-3 cos x)/(sin^(2) x)dx`

Answer

To solve the integral \(\int \frac{2 - 3 \cos x}{\sin^2 x} \, dx\), we can break it down into simpler parts. Here’s a step-by-step solution:

### Step 1: Split the Integral
We can rewrite the integral as:
\[ \int \frac{2}{\sin^2 x} \, dx - \int \frac{3 \cos x}{\sin^2 x} \, dx \]

### Step 2: Rewrite the Terms
Using the identity \(\frac{1}{\sin^2 x} = \csc^2 x\) and \(\frac{\cos x}{\sin^2 x} = \cot x \csc x\), we can express the integrals as:
\[ \int 2 \csc^2 x \, dx - 3 \int \cot x \csc x \, dx \]

### Step 3: Integrate the First Term
The integral of \(\csc^2 x\) is:
\[ \int \csc^2 x \, dx = -\cot x \]
Thus,
\[ \int 2 \csc^2 x \, dx = 2(-\cot x) = -2 \cot x \]

### Step 4: Integrate the Second Term
The integral of \(\cot x \csc x\) is:
\[ \int \cot x \csc x \, dx = -\csc x \]
Thus,
\[ -3 \int \cot x \csc x \, dx = -3(-\csc x) = 3 \csc x \]

### Step 5: Combine the Results
Now, we can combine the results from both integrals:
\[ -2 \cot x + 3 \csc x + C \]
where \(C\) is the constant of integration.

### Final Answer
The final result of the integral is:
\[ \int \frac{2 - 3 \cos x}{\sin^2 x} \, dx = 3 \csc x - 2 \cot x + C \]