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 start with the integral:
\[ I = \int_0^{\frac{\pi}{4}} \sin x \, dx \]

### Step 2: Find the antiderivative of \( \sin x \)
The antiderivative (or integral) of \( \sin x \) is:
\[ -\cos x \]

### Step 3: Apply the limits of integration
Now we apply the limits from \( 0 \) to \( \frac{\pi}{4} \):
\[ I = \left[-\cos x\right]_0^{\frac{\pi}{4}} = -\cos\left(\frac{\pi}{4}\right) - \left(-\cos(0)\right) \]
This simplifies to:
\[ I = -\cos\left(\frac{\pi}{4}\right) + \cos(0) \]

### Step 4: Evaluate \( \cos\left(\frac{\pi}{4}\right) \) and \( \cos(0) \)
We know that:
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \quad \text{and} \quad \cos(0) = 1 \]
Substituting these values into the equation gives:
\[ I = -\frac{1}{\sqrt{2}} + 1 \]

### Step 5: Simplify the expression
Now, we can combine the terms:
\[ I = 1 - \frac{1}{\sqrt{2}} \]
To express this in a single fraction, we can write:
\[ I = \frac{\sqrt{2}}{\sqrt{2}} - \frac{1}{\sqrt{2}} = \frac{\sqrt{2} - 1}{\sqrt{2}} \]

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

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