If `f(x) = sin x+cos x` and `g(x) = {:{((|x|)/(x),","x ne0),(2,","x=0):}` then the value of `int_(-pi//4)^(2pi) gof(x) dx` is equal to (a) `(3pi)/(4)` (b) `(pi)/(4)` (c) `pi` (d) None of these

Options

A. `(3pi)/(4)`
B. `(pi)/(4)`
C. `pi`
D. None of these

Answer

To solve the integral \( \int_{-\frac{\pi}{4}}^{2\pi} g(f(x)) \, dx \), where \( f(x) = \sin x + \cos x \) and \( g(x) = \frac{|x|}{x} \) for \( x \neq 0 \) and \( g(0) = 2 \), we will follow these steps:

### Step 1: Analyze the function \( f(x) \)

The function \( f(x) = \sin x + \cos x \) can be rewritten using the identity:
\[ f(x) = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]
This is because:
\[ \sin x + \cos x = \sqrt{2} \left(\frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x\right) = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]

### Step 2: Determine the range of \( f(x) \)

The maximum value of \( f(x) \) occurs when \( \sin\left(x + \frac{\pi}{4}\right) = 1 \), which gives:
\[ f(x)_{\text{max}} = \sqrt{2} \]
The minimum value occurs when \( \sin\left(x + \frac{\pi}{4}\right) = -1 \), giving:
\[ f(x)_{\text{min}} = -\sqrt{2} \]

### Step 3: Evaluate \( g(f(x)) \)

We need to evaluate \( g(f(x)) \):
- For \( f(x) > 0 \) (i.e., \( 0 < f(x) \leq \sqrt{2} \)), \( g(f(x)) = 1 \).
- For \( f(x) < 0 \) (i.e., \( -\sqrt{2} \leq f(x) < 0 \)), \( g(f(x)) = -1 \).
- At \( f(x) = 0 \), \( g(f(x)) = 2 \).

### Step 4: Find the intervals where \( f(x) \) is positive or negative

To find where \( f(x) = 0 \):
\[ \sin x + \cos x = 0 \implies \tan x = -1 \implies x = -\frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]
The solutions in the interval \( \left[-\frac{\pi}{4}, 2\pi\right] \) are:
- \( x = -\frac{\pi}{4} \)
- \( x = \frac{3\pi}{4} \)
- \( x = \frac{7\pi}{4} \)

### Step 5: Set up the integral

Now we can split the integral based on the intervals determined:
\[ \int_{-\frac{\pi}{4}}^{2\pi} g(f(x)) \, dx = \int_{-\frac{\pi}{4}}^{\frac{3\pi}{4}} (-1) \, dx + \int_{\frac{3\pi}{4}}^{\frac{7\pi}{4}} (1) \, dx + \int_{\frac{7\pi}{4}}^{2\pi} (1) \, dx \]

### Step 6: Calculate each integral

1. **First integral**:
\[ \int_{-\frac{\pi}{4}}^{\frac{3\pi}{4}} (-1) \, dx = -\left(\frac{3\pi}{4} + \frac{\pi}{4}\right) = -\pi \]

2. **Second integral**:
\[ \int_{\frac{3\pi}{4}}^{\frac{7\pi}{4}} (1) \, dx = \left(\frac{7\pi}{4} - \frac{3\pi}{4}\right) = \pi \]

3. **Third integral**:
\[ \int_{\frac{7\pi}{4}}^{2\pi} (1) \, dx = \left(2\pi - \frac{7\pi}{4}\right) = \frac{1\pi}{4} \]

### Step 7: Combine the results

Now, we combine the results of the integrals:
\[ \int_{-\frac{\pi}{4}}^{2\pi} g(f(x)) \, dx = -\pi + \pi + \frac{\pi}{4} = \frac{\pi}{4} \]

### Final Answer

Thus, the value of the integral is:
\[ \boxed{\frac{\pi}{4}} \]