`int(1)/(sin x + cos x +sqrt(2))dx` equals

Options

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

Explanation

To solve the integral \( \int \frac{1}{\sin x + \cos x + \sqrt{2}} \, dx \), we can follow these steps:

### Step 1: Simplify the Denominator
We start by rewriting the denominator:
\[ \sin x + \cos x + \sqrt{2} \]
We can factor out \( \sqrt{2} \) from the terms involving sine and cosine:
\[ = \sqrt{2} \left( \frac{\sin x}{\sqrt{2}} + \frac{\cos x}{\sqrt{2}} + 1 \right) \]
This allows us to rewrite the integral:
\[ \int \frac{1}{\sin x + \cos x + \sqrt{2}} \, dx = \int \frac{1}{\sqrt{2}} \cdot \frac{1}{\frac{\sin x}{\sqrt{2}} + \frac{\cos x}{\sqrt{2}} + 1} \, dx \]

### Step 2: Use Trigonometric Identities
Notice that \( \frac{\sin x}{\sqrt{2}} = \sin\left(x + \frac{\pi}{4}\right) \) and \( \frac{\cos x}{\sqrt{2}} = \cos\left(x + \frac{\pi}{4}\right) \). Therefore, we can rewrite:
\[ \frac{\sin x}{\sqrt{2}} + \frac{\cos x}{\sqrt{2}} = \sin\left(x + \frac{\pi}{4}\right) \]
Thus, our integral becomes:
\[ \int \frac{1}{\sqrt{2}} \cdot \frac{1}{\sin\left(x + \frac{\pi}{4}\right) + 1} \, dx \]

### Step 3: Substitute and Simplify
Now, we can use the identity \( 1 - \sin\left(x + \frac{\pi}{4}\right) = \frac{1 - \sin\left(x + \frac{\pi}{4}\right)}{1 - \sin\left(x + \frac{\pi}{4}\right)} \):
\[ \int \frac{1}{\sqrt{2}} \cdot \frac{1 - \sin\left(x + \frac{\pi}{4}\right)}{(1 - \sin\left(x + \frac{\pi}{4}\right))} \, dx \]

### Step 4: Use the Integral of \( \frac{1}{1 - \sin x} \)
The integral \( \int \frac{1}{1 - \sin x} \, dx \) can be solved using the identity:
\[ 1 - \sin x = \frac{1 - \sin x}{\cos^2\left(\frac{x}{2}\right)} \]
Thus, we can rewrite our integral:
\[ \int \frac{1}{\sqrt{2}} \cdot \frac{1}{1 - \sin\left(x + \frac{\pi}{4}\right)} \, dx \]

### Step 5: Solve the Integral
Using the integral formula, we find:
\[ \int \frac{1}{1 - \sin x} \, dx = -\cot\left(\frac{x}{2} + \frac{\pi}{8}\right) + C \]
Thus, we have:
\[ \int \frac{1}{\sqrt{2}} \cdot \frac{1}{1 - \sin\left(x + \frac{\pi}{4}\right)} \, dx = -\frac{1}{\sqrt{2}} \cot\left(\frac{x + \frac{\pi}{4}}{2}\right) + C \]

### Final Answer
Therefore, the final answer is:
\[ -\frac{1}{\sqrt{2}} \cot\left(\frac{x}{2} + \frac{\pi}{8}\right) + C \]