Evaluate the following integrals:
`int(dx)/((cosx-sinx))`

Explanation

To evaluate the integral \( \int \frac{dx}{\cos x - \sin x} \), we can follow these steps:

### Step 1: Rewrite the integral
We start with the integral:
\[ I = \int \frac{dx}{\cos x - \sin x} \]

### Step 2: Factor out \( \frac{1}{\sqrt{2}} \)
We can multiply and divide the denominator by \( \sqrt{2} \):
\[ I = \int \frac{1}{\sqrt{2}} \cdot \frac{dx}{\frac{\cos x}{\sqrt{2}} - \frac{\sin x}{\sqrt{2}}} \]

### Step 3: Recognize trigonometric identities
Notice that \( \frac{\cos x}{\sqrt{2}} \) and \( \frac{\sin x}{\sqrt{2}} \) can be expressed in terms of sine and cosine of \( 45^\circ \) (or \( \frac{\pi}{4} \)):
\[ \frac{\cos x}{\sqrt{2}} = \cos\left(x + \frac{\pi}{4}\right) \quad \text{and} \quad \frac{\sin x}{\sqrt{2}} = \sin\left(x + \frac{\pi}{4}\right) \]
Thus, we can rewrite the integral as:
\[ I = \frac{1}{\sqrt{2}} \int \frac{dx}{\cos\left(x + \frac{\pi}{4}\right) - \sin\left(x + \frac{\pi}{4}\right)} \]

### Step 4: Use a substitution
Let \( u = x + \frac{\pi}{4} \), then \( du = dx \). The integral becomes:
\[ I = \frac{1}{\sqrt{2}} \int \frac{du}{\cos u - \sin u} \]

### Step 5: Simplify the denominator
We can rewrite \( \cos u - \sin u \) using the identity:
\[ \cos u - \sin u = \sqrt{2} \left( \cos u \cos \frac{\pi}{4} - \sin u \sin \frac{\pi}{4} \right) = \sqrt{2} \cos\left(u + \frac{\pi}{4}\right) \]
Thus, we have:
\[ I = \frac{1}{\sqrt{2}} \int \frac{du}{\sqrt{2} \cos\left(u + \frac{\pi}{4}\right)} = \frac{1}{2} \int \sec\left(u + \frac{\pi}{4}\right) du \]

### Step 6: Integrate
The integral of \( \sec \) is:
\[ \int \sec x \, dx = \ln |\sec x + \tan x| + C \]
Applying this to our integral:
\[ I = \frac{1}{2} \ln \left| \sec\left(u + \frac{\pi}{4}\right) + \tan\left(u + \frac{\pi}{4}\right) \right| + C \]

### Step 7: Substitute back for \( u \)
Substituting \( u = x + \frac{\pi}{4} \):
\[ I = \frac{1}{2} \ln \left| \sec\left(x + \frac{\pi}{4}\right) + \tan\left(x + \frac{\pi}{4}\right) \right| + C \]

### Final Answer
Thus, the final result for the integral is:
\[ \int \frac{dx}{\cos x - \sin x} = \frac{1}{2} \ln \left| \sec\left(x + \frac{\pi}{4}\right) + \tan\left(x + \frac{\pi}{4}\right) \right| + C \]