Evaluate: `underset(0)overset(pi//4)intsinxdx`

Explanation

To evaluate the integral \(\int_0^{\frac{\pi}{4}} \sin x \, dx\), we will follow these steps:

### Step 1: Set up the integral
We need to evaluate the definite integral of \(\sin x\) from \(0\) to \(\frac{\pi}{4}\):
\[ \int_0^{\frac{\pi}{4}} \sin x \, dx \]

### Step 2: Find the indefinite integral of \(\sin x\)
The integral of \(\sin x\) is:
\[ -\cos x + C \]
where \(C\) is the constant of integration.

### Step 3: Apply the limits of integration
Now, we will evaluate the definite integral by applying the limits \(0\) and \(\frac{\pi}{4}\):
\[ \left[-\cos x\right]_0^{\frac{\pi}{4}} = -\cos\left(\frac{\pi}{4}\right) - \left(-\cos(0)\right) \]

### Step 4: Calculate \(\cos\left(\frac{\pi}{4}\right)\) and \(\cos(0)\)
We know that:
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \quad \text{and} \quad \cos(0) = 1 \]

### Step 5: Substitute the values into the expression
Substituting these values back into the expression, we get:
\[ -\left(\frac{1}{\sqrt{2}}\right) + 1 = 1 - \frac{1}{\sqrt{2}} \]

### Step 6: Simplify the expression
To simplify \(1 - \frac{1}{\sqrt{2}}\), we can convert \(1\) into a fraction with a common denominator:
\[ 1 = \frac{\sqrt{2}}{\sqrt{2}} \quad \Rightarrow \quad 1 - \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2}} - \frac{1}{\sqrt{2}} = \frac{\sqrt{2} - 1}{\sqrt{2}} \]

### Final Answer
Thus, the value of the integral \(\int_0^{\frac{\pi}{4}} \sin x \, dx\) is:
\[ \frac{\sqrt{2} - 1}{\sqrt{2}} \]