Evaluate :`int_(-pi//4)^(pi//4) |sin x|dx`

Answer

To evaluate the integral \( \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} |\sin x| \, dx \), we can follow these steps:

### Step 1: Understand the function \( |\sin x| \)
The function \( |\sin x| \) is the absolute value of \( \sin x \). We need to analyze the behavior of \( \sin x \) in the interval \( \left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \).

### Step 2: Determine where \( \sin x \) is positive or negative
In the interval \( \left[-\frac{\pi}{4}, \frac{\pi}{4}\right] \):
- \( \sin x \) is positive for \( x \) in \( \left[0, \frac{\pi}{4}\right] \).
- \( \sin x \) is negative for \( x \) in \( \left[-\frac{\pi}{4}, 0\right] \).

Thus, we can rewrite the integral as:
\[ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} |\sin x| \, dx = \int_{-\frac{\pi}{4}}^{0} -\sin x \, dx + \int_{0}^{\frac{\pi}{4}} \sin x \, dx \]

### Step 3: Evaluate the first integral
For the first integral:
\[ \int_{-\frac{\pi}{4}}^{0} -\sin x \, dx = -\left[-\cos x\right]_{-\frac{\pi}{4}}^{0} = -\left[-\cos(0) + \cos\left(-\frac{\pi}{4}\right)\right] \]
Calculating the values:
- \( \cos(0) = 1 \)
- \( \cos\left(-\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \)

Thus, we have:
\[ -\left[-1 + \frac{1}{\sqrt{2}}\right] = 1 - \frac{1}{\sqrt{2}} \]

### Step 4: Evaluate the second integral
For the second integral:
\[ \int_{0}^{\frac{\pi}{4}} \sin x \, dx = -\left[\cos x\right]_{0}^{\frac{\pi}{4}} = -\left[\cos\left(\frac{\pi}{4}\right) - \cos(0)\right] \]
Calculating the values:
- \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \)
- \( \cos(0) = 1 \)

Thus, we have:
\[ -\left[\frac{1}{\sqrt{2}} - 1\right] = 1 - \frac{1}{\sqrt{2}} \]

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

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