`int_(0)^(pi/2) |sinx - cosx |dx` is equal to

Options

A. `0`
B. `2(sqrt(2) -1)`
C. `2sqrt(2)`
D. `2(sqrt(2) + 1)`

Explanation

To solve the integral \( \int_{0}^{\frac{\pi}{2}} |\sin x - \cos x| \, dx \), we will follow these steps:

### Step 1: Determine where \( \sin x - \cos x \) changes sign
We need to find the points where \( \sin x = \cos x \). This occurs when:
\[ \tan x = 1 \quad \Rightarrow \quad x = \frac{\pi}{4} \]
Thus, we have two intervals to consider:
1. From \( 0 \) to \( \frac{\pi}{4} \)
2. From \( \frac{\pi}{4} \) to \( \frac{\pi}{2} \)

### Step 2: Analyze the sign of \( \sin x - \cos x \)
- For \( x \in [0, \frac{\pi}{4}) \), \( \sin x < \cos x \) so \( \sin x - \cos x < 0 \) and \( |\sin x - \cos x| = -(\sin x - \cos x) = \cos x - \sin x \).
- For \( x \in (\frac{\pi}{4}, \frac{\pi}{2}] \), \( \sin x > \cos x \) so \( \sin x - \cos x > 0 \) and \( |\sin x - \cos x| = \sin x - \cos x \).

### Step 3: Set up the integral
We can split the integral into two parts:
\[ \int_{0}^{\frac{\pi}{2}} |\sin x - \cos x| \, dx = \int_{0}^{\frac{\pi}{4}} (\cos x - \sin x) \, dx + \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (\sin x - \cos x) \, dx \]

### Step 4: Evaluate the first integral
Calculate \( \int_{0}^{\frac{\pi}{4}} (\cos x - \sin x) \, dx \):
\[ \int (\cos x - \sin x) \, dx = \sin x + \cos x \]
Evaluating from \( 0 \) to \( \frac{\pi}{4} \):
\[ \left[ \sin x + \cos x \right]_{0}^{\frac{\pi}{4}} = \left( \sin \frac{\pi}{4} + \cos \frac{\pi}{4} \right) - \left( \sin 0 + \cos 0 \right) = \left( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \right) - (0 + 1) = \sqrt{2} - 1 \]

### Step 5: Evaluate the second integral
Calculate \( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (\sin x - \cos x) \, dx \):
\[ \int (\sin x - \cos x) \, dx = -\cos x - \sin x \]
Evaluating from \( \frac{\pi}{4} \) to \( \frac{\pi}{2} \):
\[ \left[ -\cos x - \sin x \right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} = \left( -\cos \frac{\pi}{2} - \sin \frac{\pi}{2} \right) - \left( -\cos \frac{\pi}{4} - \sin \frac{\pi}{4} \right) = \left( 0 - 1 \right) - \left( -\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} \right) = -1 + \sqrt{2} \]

### Step 6: Combine the results
Now, we combine both parts:
\[ \int_{0}^{\frac{\pi}{2}} |\sin x - \cos x| \, dx = (\sqrt{2} - 1) + (-1 + \sqrt{2}) = 2\sqrt{2} - 2 \]

### Final Result
Thus, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{2}} |\sin x - \cos x| \, dx = 2(\sqrt{2} - 1) \]