The value of `sqrt(2)int(sinx)/(sin(x-(pi)/(4)))dx` , is

Options

A. `x+log|cos (x-pi/4)|+C`
B. `x+log|sin (x-pi/4)|+C`
C. `x-log|sin (x- pi/4)|+C`
D. `x-log |cos (x- pi/4)|+C`

Answer

To solve the integral \( \sqrt{2} \int \frac{\sin x}{\sin(x - \frac{\pi}{4})} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We can express the integral as:
\[ I = \sqrt{2} \int \frac{\sin x}{\sin(x - \frac{\pi}{4})} \, dx \]
We can use the identity for sine, where \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

### Step 2: Apply the Sine Addition Formula
Using the identity, we rewrite \( \sin x \):
\[ \sin x = \sin\left((x - \frac{\pi}{4}) + \frac{\pi}{4}\right) = \sin(x - \frac{\pi}{4}) \cos\frac{\pi}{4} + \cos(x - \frac{\pi}{4}) \sin\frac{\pi}{4} \]
Since \( \cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{1}{\sqrt{2}} \), we have:
\[ \sin x = \frac{1}{\sqrt{2}} \sin(x - \frac{\pi}{4}) + \frac{1}{\sqrt{2}} \cos(x - \frac{\pi}{4}) \]

### Step 3: Substitute in the Integral
Substituting this back into the integral gives:
\[ I = \sqrt{2} \int \frac{\frac{1}{\sqrt{2}} \sin(x - \frac{\pi}{4}) + \frac{1}{\sqrt{2}} \cos(x - \frac{\pi}{4})}{\sin(x - \frac{\pi}{4})} \, dx \]
This simplifies to:
\[ I = \sqrt{2} \int \left(1 + \cot(x - \frac{\pi}{4})\right) \, dx \]

### Step 4: Separate the Integral
We can separate the integral:
\[ I = \sqrt{2} \left( \int 1 \, dx + \int \cot(x - \frac{\pi}{4}) \, dx \right) \]

### Step 5: Integrate Each Part
1. The integral of \( 1 \) is simply \( x \).
2. The integral of \( \cot(x - \frac{\pi}{4}) \) is:
\[ \int \cot(x - \frac{\pi}{4}) \, dx = \log|\sin(x - \frac{\pi}{4})| + C \]

### Step 6: Combine the Results
Putting it all together:
\[ I = \sqrt{2} \left( x + \log|\sin(x - \frac{\pi}{4})| \right) + C \]

### Step 7: Simplify the Expression
Since \( \sqrt{2} \) is a constant factor, we can distribute it:
\[ I = \sqrt{2} x + \sqrt{2} \log|\sin(x - \frac{\pi}{4})| + C \]

### Final Answer
Thus, the value of the integral is:
\[ \sqrt{2} x + \sqrt{2} \log|\sin(x - \frac{\pi}{4})| + C \]