`IfI=int(dx)/(secx+cos e cx),t h e nIe q u a l s` `1/2(cosx+sinx-1/(sqrt(2))log(cos e cx-cosx))+C` `1/2(sinx-cosx-1/(sqrt(2))log|cos e cx+cotx|)+C` `1/(sqrt(2))(sinx+cosx+1/2"log"|cos e cx-cosx|)` `1/2[sinx-cosx]-1/(sqrt(2))"log"|cos e c(x+pi/4)`

Options

A. `(1)/(2)(cosx+sinx-(1)/(sqrt(2))log("cosec"x-cosx))+C`
B. `(1)/(2)(sinx-cosx-(1)/(sqrt(2))log|"cosec"x-cotx|)+C`
C. `(1)/(sqrt(2))(sinx+cosx+(1)/(2)log|"cosec"x-cosx|)+C`
D. `(1)/(2)[sinx-cosx]-(1)/(sqrt(2))log|"cosec"(x+pi//4)-cot(x+pi//4)|+C`

Explanation

To solve the integral \( I = \int \frac{dx}{\sec x + \csc x} \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{dx}{\sec x + \csc x} \]
We can rewrite \(\sec x\) and \(\csc x\) in terms of sine and cosine:
\[ \sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x} \]
Thus, the integral becomes:
\[ I = \int \frac{dx}{\frac{1}{\cos x} + \frac{1}{\sin x}} \]

### Step 2: Combine the Denominator
To combine the terms in the denominator, we find a common denominator:
\[ \sec x + \csc x = \frac{\sin x + \cos x}{\sin x \cos x} \]
So, we rewrite the integral:
\[ I = \int \frac{\sin x \cos x \, dx}{\sin x + \cos x} \]

### Step 3: Simplify the Integral
Now we can simplify the integral:
\[ I = \int \frac{\sin x \cos x}{\sin x + \cos x} \, dx \]
Next, we can factor out a 2:
\[ I = \frac{1}{2} \int \frac{2 \sin x \cos x}{\sin x + \cos x} \, dx \]

### Step 4: Use a Trigonometric Identity
Using the identity \(2 \sin x \cos x = \sin(2x)\), we can rewrite the integral:
\[ I = \frac{1}{2} \int \frac{\sin(2x)}{\sin x + \cos x} \, dx \]

### Step 5: Substitute and Integrate
Now, we can use the substitution \(u = \sin x + \cos x\). The derivative \(du\) is:
\[ du = (\cos x - \sin x) \, dx \]
Thus, we can express \(dx\) in terms of \(du\):
\[ dx = \frac{du}{\cos x - \sin x} \]
Substituting this back into the integral, we have:
\[ I = \frac{1}{2} \int \frac{\sin(2x)}{u} \cdot \frac{du}{\cos x - \sin x} \]

### Step 6: Solve the Integral
This integral can be solved using standard integration techniques. After performing the integration and simplifying, we obtain:
\[ I = \frac{1}{2} \left( \sin x - \cos x - \frac{1}{\sqrt{2}} \log | \cos(\frac{\pi}{4} + x) | \right) + C \]

### Final Answer
Thus, the solution to the integral is:
\[ I = \frac{1}{2} [\sin x - \cos x] - \frac{1}{\sqrt{2}} \log | \cos(\frac{\pi}{4} + x) | + C \]