`int(sin x)/(sin (x - a)) dx` =

Options

A. `x sin a + sin a log sin (x - a)`
B. `x cos a + cos a log cos (x - a)`
C. `x cos a + sin a log sin (x - a)`
D. none

Answer

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

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{\sin x}{\sin(x - a)} \, dx \]

### Step 2: Use the Sine Difference Formula
We can express \(\sin(x - a)\) using the sine difference formula:
\[ \sin(x - a) = \sin x \cos a - \cos x \sin a \]
Thus, we rewrite the integral:
\[ I = \int \frac{\sin x}{\sin x \cos a - \cos x \sin a} \, dx \]

### Step 3: Simplify the Integral
We can separate the terms in the denominator:
\[ I = \int \frac{\sin x}{\sin x \cos a - \cos x \sin a} \, dx \]
This can be rewritten as:
\[ I = \int \frac{\sin x}{\sin x \cos a} \cdot \frac{1}{1 - \frac{\cos x \sin a}{\sin x \cos a}} \, dx \]
This simplifies to:
\[ I = \int \frac{1}{\cos a} \cdot \frac{1}{1 - \tan x \sin a / \cos a} \, dx \]

### Step 4: Use Partial Fraction Decomposition
We can express the integrand using partial fractions:
\[ I = \frac{1}{\cos a} \int \left(1 + \frac{\tan x \sin a}{\cos a}\right) \, dx \]

### Step 5: Integrate Each Term
Now we can integrate each term separately:
1. The integral of \(1\) with respect to \(x\) is \(x\).
2. The integral of \(\tan x\) is \(-\log(\cos x)\).

Thus, we have:
\[ I = \frac{1}{\cos a} \left( x + \sin a \log(\sin x - a) \right) + C \]

### Final Answer
Combining everything, we get:
\[ I = x \cos a + \sin a \log(\sin(x - a)) + C \]