Evaluate: `int(sin(a-x))/(sinx)dx`

Answer

To evaluate the integral \(\int \frac{\sin(a - x)}{\sin x} \, dx\), we can follow these steps:

### Step 1: Use the sine subtraction formula
We start by applying the sine subtraction formula:
\[ \sin(a - x) = \sin a \cos x - \cos a \sin x. \]
Thus, we can rewrite the integral as:
\[ \int \frac{\sin(a - x)}{\sin x} \, dx = \int \frac{\sin a \cos x - \cos a \sin x}{\sin x} \, dx. \]

### Step 2: Split the integral
Now, we can split the integral into two separate integrals:
\[ \int \frac{\sin a \cos x}{\sin x} \, dx - \int \frac{\cos a \sin x}{\sin x} \, dx. \]
This simplifies to:
\[ \int \sin a \cot x \, dx - \int \cos a \, dx. \]

### Step 3: Factor out constants
Since \(\sin a\) and \(\cos a\) are constants with respect to \(x\), we can factor them out of the integrals:
\[ \sin a \int \cot x \, dx - \cos a \int 1 \, dx. \]

### Step 4: Evaluate the integrals
Now, we evaluate the integrals:
1. The integral of \(\cot x\) is \(\log |\sin x|\).
2. The integral of \(1\) is \(x\).

Thus, we have:
\[ \sin a \log |\sin x| - \cos a \cdot x + C, \]
where \(C\) is the constant of integration.

### Final Answer
The final result is:
\[ \int \frac{\sin(a - x)}{\sin x} \, dx = \sin a \log |\sin x| - \cos a \cdot x + C. \]

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