`int(1)/(cosx-sinx)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)-(3pi)/(8))|+C`
D. `(1)/(sqrt(2))log|tan((x)/(2)-(pi)/(8))|+C`

Answer

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

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

We start by multiplying and dividing the integrand by \( \frac{1}{\sqrt{2}} \):

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

### Step 2: Rewrite the Integral

This can be rewritten as:

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

### Step 3: Use Trigonometric Identities

Using the fact that \( \frac{1}{\sqrt{2}} = \cos \frac{\pi}{4} \) and \( \frac{1}{\sqrt{2}} = \sin \frac{\pi}{4} \), we can express the integral as:

\[ I = \frac{1}{\sqrt{2}} \int \frac{1}{\cos \frac{\pi}{4} \cos x - \sin \frac{\pi}{4} \sin x} \, dx \]

### Step 4: Apply the Cosine Addition Formula

Using the cosine addition formula \( \cos(a + b) = \cos a \cos b - \sin a \sin b \):

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

### Step 5: Rewrite the Integral in Terms of Secant

This can be rewritten as:

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

### Step 6: Integrate Secant

The integral of secant is known:

\[ \int \sec x \, dx = \ln |\sec x + \tan x| + C \]

Thus, we have:

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

### Step 7: Substitute Back

Now, substituting 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:

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