Evaluate the following :
`int_(0)^(pi//2)sin theta d theta`

Answer

To evaluate the integral \( \int_{0}^{\frac{\pi}{2}} \sin \theta \, d\theta \), we will follow these steps:

### Step 1: Set up the integral
We begin by writing the integral we need to evaluate:
\[ I = \int_{0}^{\frac{\pi}{2}} \sin \theta \, d\theta \]

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

### Step 3: Evaluate the definite integral
Now we will evaluate the definite integral from \(0\) to \(\frac{\pi}{2}\):
\[ I = \left[-\cos \theta\right]_{0}^{\frac{\pi}{2}} \]

### Step 4: Substitute the limits of integration
Now we substitute the upper limit and lower limit into the antiderivative:
\[ I = -\cos\left(\frac{\pi}{2}\right) - \left(-\cos(0)\right) \]

### Step 5: Calculate the cosine values
We know that:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad \cos(0) = 1 \]
Substituting these values gives:
\[ I = -0 - (-1) = 1 \]

### Final Result
Thus, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{2}} \sin \theta \, d\theta = 1 \]