If `int(sinx)/(sin(x-(pi)/(4)))dx=Af(x)+(1)/(sqrt2)log[|sinx-cosx|]+c,` then

Options

A. `A=(1)/(sqrt2),f(x)=sinx`
B. `A=sqrt2,f(x)=cosx`
C. `A=sqrt2,f(x)=x`
D. `A=(1)/(sqrt2),f(x)=x`

Explanation

To solve the integral \(\int \frac{\sin x}{\sin\left(x - \frac{\pi}{4}\right)} \, dx\), we will follow these steps:

### Step 1: Rewrite the numerator
We can express \(\sin x\) using the angle addition formula:
\[ \sin x = \sin\left(x - \frac{\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right) \cos\left(x - \frac{\pi}{4}\right) \]
This allows us to rewrite the integral:
\[ I = \int \frac{\sin\left(x - \frac{\pi}{4}\right) + \frac{1}{\sqrt{2}} \cos\left(x - \frac{\pi}{4}\right)}{\sin\left(x - \frac{\pi}{4}\right)} \, dx \]

### Step 2: Split the integral
Now we can split the integral into two parts:
\[ I = \int 1 \, dx + \frac{1}{\sqrt{2}} \int \cot\left(x - \frac{\pi}{4}\right) \, dx \]

### Step 3: Integrate the first part
The first integral is straightforward:
\[ \int 1 \, dx = x \]

### Step 4: Integrate the second part
For the second integral, we know that:
\[ \int \cot u \, du = \log|\sin u| + C \]
Thus,
\[ \int \cot\left(x - \frac{\pi}{4}\right) \, dx = \log|\sin\left(x - \frac{\pi}{4}\right)| + C \]

### Step 5: Combine the results
Combining both parts, we have:
\[ I = x + \frac{1}{\sqrt{2}} \log|\sin\left(x - \frac{\pi}{4}\right)| + C \]

### Step 6: Rewrite the logarithm
Using the identity \(\sin\left(x - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}(\sin x - \cos x)\), we can rewrite the logarithm:
\[ I = x + \frac{1}{\sqrt{2}} \log\left|\frac{1}{\sqrt{2}}(\sin x - \cos x)\right| + C \]
This simplifies to:
\[ I = x + \frac{1}{\sqrt{2}} \log|\sin x - \cos x| - \frac{1}{\sqrt{2}} \log\left(\sqrt{2}\right) + C \]

### Step 7: Final expression
Thus, we can express the integral as:
\[ I = x + \frac{1}{\sqrt{2}} \log|\sin x - \cos x| + C' \]
where \(C' = C - \frac{1}{\sqrt{2}} \log\left(\sqrt{2}\right)\).

### Conclusion
From the given expression, we can identify:
\[ A = \frac{1}{\sqrt{2}}, \quad f(x) = x \]