`int_(0)^(pi//2) abs(sin(x-pi/4)) dx=`

Options

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

Answer

To solve the integral \( \int_{0}^{\frac{\pi}{2}} |\sin(x - \frac{\pi}{4})| \, dx \), we will first analyze the expression inside the absolute value.

### Step 1: Determine where the expression changes sign
The expression \( \sin(x - \frac{\pi}{4}) \) will change sign at the point where \( x - \frac{\pi}{4} = 0 \), which gives us \( x = \frac{\pi}{4} \).

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

### Step 3: Evaluate the first integral
For \( x \) in the interval \( [0, \frac{\pi}{4}] \), \( \sin(x - \frac{\pi}{4}) \) is negative, so we have:
\[ \int_{0}^{\frac{\pi}{4}} -\sin(x - \frac{\pi}{4}) \, dx = -\int_{0}^{\frac{\pi}{4}} \sin(x - \frac{\pi}{4}) \, dx \]
Using the substitution \( u = x - \frac{\pi}{4} \), we have \( du = dx \) and the limits change from \( x = 0 \) to \( u = -\frac{\pi}{4} \) and from \( x = \frac{\pi}{4} \) to \( u = 0 \):
\[ = -\int_{-\frac{\pi}{4}}^{0} \sin(u) \, du = -[-\cos(u)]_{-\frac{\pi}{4}}^{0} = -[-\cos(0) + \cos(-\frac{\pi}{4})] = -[-1 + \frac{1}{\sqrt{2}}] = 1 - \frac{1}{\sqrt{2}} \]

### Step 4: Evaluate the second integral
For \( x \) in the interval \( [\frac{\pi}{4}, \frac{\pi}{2}] \), \( \sin(x - \frac{\pi}{4}) \) is positive, so we have:
\[ \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \sin(x - \frac{\pi}{4}) \, dx \]
Using the same substitution \( u = x - \frac{\pi}{4} \), the limits change from \( x = \frac{\pi}{4} \) to \( u = 0 \) and from \( x = \frac{\pi}{2} \) to \( u = \frac{\pi}{4} \):
\[ = \int_{0}^{\frac{\pi}{4}} \sin(u) \, du = [-\cos(u)]_{0}^{\frac{\pi}{4}} = -\cos(\frac{\pi}{4}) + \cos(0) = -\frac{1}{\sqrt{2}} + 1 = 1 - \frac{1}{\sqrt{2}} \]

### Step 5: Combine the results
Now we can combine both parts:
\[ \int_{0}^{\frac{\pi}{2}} |\sin(x - \frac{\pi}{4})| \, dx = \left(1 - \frac{1}{\sqrt{2}}\right) + \left(1 - \frac{1}{\sqrt{2}}\right) = 2\left(1 - \frac{1}{\sqrt{2}}\right) \]

### Final Result
The final result of the integral is:
\[ \int_{0}^{\frac{\pi}{2}} |\sin(x - \frac{\pi}{4})| \, dx = 2\left(1 - \frac{1}{\sqrt{2}}\right) \]