`int(dx)/(sinx-cosx)=`

Options

A. `(1)/(sqrt2)log tan((x)/(2)-(pi)/(8))+c`
B. `logtan((x)/(2)+(pi)/(8))+c`
C. `(1)/(2)logtan((x)/(2)+(pi)/(8))+c`
D. `(1)/(sqrt2)log tan(x+(pi)/(4))+c`

Answer

To solve the integral \(\int \frac{dx}{\sin x - \cos x}\), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Multiply and Divide by \(\sqrt{2}\)**:
We start by multiplying the denominator by \(\sqrt{2}\) and dividing by \(\sqrt{2}\) to simplify the expression:
\[ I = \int \frac{dx}{\sin x - \cos x} = \int \frac{\sqrt{2} \, dx}{\sqrt{2}(\sin x - \cos x)} = \frac{1}{\sqrt{2}} \int \frac{dx}{\frac{1}{\sqrt{2}} \sin x - \frac{1}{\sqrt{2}} \cos x} \]

2. **Recognize Trigonometric Values**:
We know that \(\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\). Thus, we can rewrite the integral:
\[ I = \frac{1}{\sqrt{2}} \int \frac{dx}{\sin x - \cos x} = \frac{1}{\sqrt{2}} \int \frac{dx}{\sin x - \sin \frac{\pi}{4} \cos x} \]

3. **Use the Sine Difference Formula**:
We can express \(\sin x - \cos x\) in terms of a sine function:
\[ \sin x - \cos x = \sqrt{2} \left( \sin x \cos \frac{\pi}{4} - \cos x \sin \frac{\pi}{4} \right) = \sqrt{2} \sin\left(x - \frac{\pi}{4}\right) \]
Therefore, we can rewrite the integral:
\[ I = \frac{1}{\sqrt{2}} \int \frac{dx}{\sqrt{2} \sin\left(x - \frac{\pi}{4}\right)} = \frac{1}{2} \int \frac{dx}{\sin\left(x - \frac{\pi}{4}\right)} \]

4. **Integrate Using Cosecant**:
The integral of \(\csc u\) is \(-\ln |\csc u + \cot u| + C\). Thus:
\[ I = \frac{1}{2} \left(-\ln |\csc\left(x - \frac{\pi}{4}\right) + \cot\left(x - \frac{\pi}{4}\right)|\right) + C \]

5. **Express in Terms of Tangent**:
We can express \(\csc\) and \(\cot\) in terms of tangent:
\[ \csc\left(x - \frac{\pi}{4}\right) = \frac{1}{\sin\left(x - \frac{\pi}{4}\right)}, \quad \cot\left(x - \frac{\pi}{4}\right) = \frac{\cos\left(x - \frac{\pi}{4}\right)}{\sin\left(x - \frac{\pi}{4}\right)} \]
Thus, we can rewrite the integral as:
\[ I = \frac{1}{2} \left(-\ln \left|\frac{1 + \cos\left(x - \frac{\pi}{4}\right)}{\sin\left(x - \frac{\pi}{4}\right)}\right|\right) + C \]

6. **Final Result**:
After simplifying, we arrive at the final result:
\[ I = \frac{1}{\sqrt{2}} \ln \left|\tan\left(\frac{x}{2} - \frac{\pi}{8}\right)\right| + C \]

### Final Answer:
\[ \int \frac{dx}{\sin x - \cos x} = \frac{1}{\sqrt{2}} \ln \left|\tan\left(\frac{x}{2} - \frac{\pi}{8}\right)\right| + C \]