`int (1)/(cos x-sinx )dx=`…

Options

A. `(1)/(sqrt2)log ["cosec"(x+(pi)/(4))-cot (x+(pi)/(4))]+c`
B. `sqrt2 log [cosec (x+(pi)/(4))+cot (x+(pi)/(4))]+c`
C. `(1)/(sqrt2)log [sec (x+(pi)/(4)) + tan (x+(pi)/(4))]+c`
D. `sqrt2 log [sec (x+(pi)/(4))-tan (x+(pi)/(4))]+c`

Answer

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

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

### Step 2: Multiply by \(\frac{1}{\sqrt{2}}\)
To simplify the expression, we multiply the numerator and denominator by \(\frac{1}{\sqrt{2}}\):
\[ I = \int \frac{1/\sqrt{2}}{(1/\sqrt{2})\cos x - (1/\sqrt{2})\sin x} \, dx \]

### Step 3: Recognize the Trigonometric Identity
We can rewrite the denominator using the angle addition formula:
\[ \frac{1}{\sqrt{2}} \cos x - \frac{1}{\sqrt{2}} \sin x = \cos\left(\frac{\pi}{4}\right) \cos x - \sin\left(\frac{\pi}{4}\right) \sin x = \cos\left(x + \frac{\pi}{4}\right) \]
Thus, we have:
\[ I = \int \frac{1/\sqrt{2}}{\cos\left(x + \frac{\pi}{4}\right)} \, dx \]

### Step 4: Factor out \(\frac{1}{\sqrt{2}}\)
Now, factor out \(\frac{1}{\sqrt{2}}\):
\[ I = \frac{1}{\sqrt{2}} \int \sec\left(x + \frac{\pi}{4}\right) \, dx \]

### Step 5: Integrate the Secant Function
The integral of \(\sec u\) is:
\[ \int \sec u \, du = \ln |\sec u + \tan u| + C \]
Applying this to our integral:
\[ I = \frac{1}{\sqrt{2}} \left( \ln \left| \sec\left(x + \frac{\pi}{4}\right) + \tan\left(x + \frac{\pi}{4}\right) \right| + C \right) \]

### Step 6: Final Expression
Thus, we can express the final answer as:
\[ I = \frac{1}{\sqrt{2}} \ln \left| \sec\left(x + \frac{\pi}{4}\right) + \tan\left(x + \frac{\pi}{4}\right) \right| + C \]

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