To solve the integral of \( \frac{1}{\sin x - \cos x} \, dx \), we can use a trigonometric identity and substitution method.

### Step-by-Step Solution:

1. **Simplify the expression**:
   \[
   I = \int \frac{1}{\sin x - \cos x} \, dx
   \]
   Multiply the numerator and denominator by \( \frac{1}{\sin x + \cos x} \) (which is the conjugate of the denominator):
   \[
   I = \int \frac{\sin x + \cos x}{(\sin x - \cos x)(\sin x + \cos x)} \, dx
   \]
   Simplify the denominator using the identity \( a^2 - b^2 = (a - b)(a + b) \):
   \[
   I = \int \frac{\sin x + \cos x}{\sin^2 x - \cos^2 x} \, dx
   \]
   Use the identity \( \sin^2 x - \cos^2 x = -\cos(2x) \):
   \[
   I = -\int \frac{\sin x + \cos x}{\cos(2x)} \, dx
   \]

2. **Substitute**:
   Let \( u = \sin x + \cos x \), then \( du = (\cos x - \sin x) \, dx \).

   Notice that the integral can be split:
   \[
   I = -\int \frac{\sin x + \cos x}{\cos(2x)} \, dx = -\int \frac{u}{\cos(2x)} \, dx
   \]
   But since we have \( du = (\cos x - \sin x) \, dx \), we can rewrite \( u^2 \) as follows:
   \[
   u^2 = \sin^2 x + \cos^2 x + 2\sin x \cos x = 1 + \sin(2x)
   \]
   So:
   \[
   I = \int \frac{u \, du}{1 - u^2}
   \]

3. **Final integration**:
   The integral can now be simplified by recognizing it as a standard form:
   \[
   I = \frac{1}{\sqrt{2}} \ln \left| \frac{\sin x + \cos x - \sqrt{2}}{\sin x + \cos x + \sqrt{2}} \right| + C
   \]

Thus, the integral of \( \frac{1}{\sin x - \cos x} \) is:
\[
\boxed{\frac{1}{\sqrt{2}} \ln \left| \frac{\sin x + \cos x - \sqrt{2}}{\sin x + \cos x + \sqrt{2}} \right| + C}
\]

Would you like more details or have any questions?

### Related Questions:
1. How can trigonometric identities simplify integrals?
2. What are common substitution techniques in trigonometry?
3. How do conjugates help in integration?
4. What are other methods for integrating trigonometric functions?
5. How do you verify an integration result?

**Tip:** Always look for symmetry or standard identities when dealing with trigonometric integrals; they often simplify the problem significantly.

Explore the integral of 1/(sin x - cos x) with a detailed step-by-step solution. Learn how trigonometric identities and substitution methods simplify the integration process. Suitable for college-level students studying calculus and trigonometry.

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