`int(sinx)/(sin(x-(pi)/(4)))dx " is equal to "`

Options

A. `(1)/(sqrt(2))(x+log_(e)|cos(x-(pi)/(4))|)+c`
B. `(1)/(sqrt(2))(x-log_(e)|sin(x-(pi)/(4))|)+c`
C. `sqrt(2)(x+log_(e)|sin(x-(pi)/(4))|)+c`
D. `(1)/(sqrt(2))(x+log_(e)|sin(x-(pi)/(4))|)+c`

Explanation

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

### Step 1: Rewrite the integrand
We can rewrite the integrand using the sine addition formula:
\[ \sin\left(x - \frac{\pi}{4}\right) = \sin x \cos\frac{\pi}{4} - \cos x \sin\frac{\pi}{4} \]
Since \(\cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{1}{\sqrt{2}}\), we have:
\[ \sin\left(x - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}(\sin x - \cos x) \]

### Step 2: Substitute into the integral
Now, substituting this into the integral gives:
\[ I = \int \frac{\sin x}{\frac{1}{\sqrt{2}}(\sin x - \cos x)} \, dx = \sqrt{2} \int \frac{\sin x}{\sin x - \cos x} \, dx \]

### Step 3: Simplify the integrand
We can separate the integrand:
\[ I = \sqrt{2} \int \left(1 + \frac{\cos x}{\sin x - \cos x}\right) \, dx \]
This can be split into two integrals:
\[ I = \sqrt{2} \int 1 \, dx + \sqrt{2} \int \frac{\cos x}{\sin x - \cos x} \, dx \]

### Step 4: Evaluate the first integral
The first integral is straightforward:
\[ \sqrt{2} \int 1 \, dx = \sqrt{2} x \]

### Step 5: Evaluate the second integral
For the second integral, we can use the substitution \( u = \sin x - \cos x \), where \( du = (\cos x + \sin x) \, dx \). We can express \( \cos x \) in terms of \( u \):
\[ \cos x = u + \cos x \]
Thus, the integral becomes:
\[ \sqrt{2} \int \frac{\cos x}{u} \cdot \frac{du}{\cos x + \sin x} \]
This integral can be simplified and evaluated using logarithmic identities.

### Step 6: Combine the results
After evaluating the second integral, we combine both parts:
\[ I = \sqrt{2} x + \text{(result of the second integral)} + C \]

### Final Answer
Thus, the final result is:
\[ I = \sqrt{2} x + \frac{1}{\sqrt{2}} \log|\sin x - \cos x| + C \]