`int(1)/(cos x - sin x )dx` is equal to

Options

A. `(1)/(sqrt(2))log|tan((x)/(2)-(3pi)/(8))|+C`
B. `(1)/(sqrt(2))log|"cot"(x)/(2)|+C`
C. `(1)/(sqrt(2))log|tan((x)/(2)-(pi)/(8))+C`
D. `(1)/(sqrt(2))log|tan((x)/(2)+(3pi)/(8))|+C`

Explanation

To solve the integral \( I = \int \frac{1}{\cos x - \sin x} \, dx \), we can follow these steps:

### Step 1: Multiply and Divide by \(\sqrt{2}\)

We start by multiplying and dividing the integrand by \(\sqrt{2}\):

\[ I = \int \frac{1}{\cos x - \sin x} \, dx = \int \frac{\sqrt{2}}{\sqrt{2}(\cos x - \sin x)} \, dx \]

### Step 2: Rewrite the Denominator

Now we can rewrite the denominator:

\[ I = \frac{1}{\sqrt{2}} \int \frac{1}{\frac{1}{\sqrt{2}} \cos x - \frac{1}{\sqrt{2}} \sin x} \, dx \]

### Step 3: Use the Cosine of a Sum Formula

Recognizing that \(\frac{1}{\sqrt{2}} \cos x - \frac{1}{\sqrt{2}} \sin x\) can be expressed as \(\cos\left(x + \frac{\pi}{4}\right)\):

\[ I = \frac{1}{\sqrt{2}} \int \frac{1}{\cos\left(x + \frac{\pi}{4}\right)} \, dx \]

### Step 4: Change of Variable

Using the identity \(\sec x = \frac{1}{\cos x}\):

\[ I = \frac{1}{\sqrt{2}} \int \sec\left(x + \frac{\pi}{4}\right) \, dx \]

### Step 5: Integrate the Secant Function

The integral of \(\sec x\) is \(\ln | \sec x + \tan x | + C\):

\[ I = \frac{1}{\sqrt{2}} \ln \left| \sec\left(x + \frac{\pi}{4}\right) + \tan\left(x + \frac{\pi}{4}\right) \right| + C \]

### Step 6: Substitute Back

Now we substitute back \(x + \frac{\pi}{4}\):

\[ I = \frac{1}{\sqrt{2}} \ln \left| \sec\left(x + \frac{\pi}{4}\right) + \tan\left(x + \frac{\pi}{4}\right) \right| + C \]

### Final Answer

Thus, the final result for the integral is:

\[ I = \frac{1}{\sqrt{2}} \ln \left| \sec\left(x + \frac{\pi}{4}\right) + \tan\left(x + \frac{\pi}{4}\right) \right| + C \]