Evaluate the following integrals:
`int sqrt(1+sinx dx`

Answer

To evaluate the integral \( I = \int \sqrt{1 + \sin x} \, dx \), we can follow these steps:

### Step 1: Simplify the integrand
We start by using the identity for \( \sin x \):
\[ 1 + \sin x = 1 + 2 \sin \frac{x}{2} \cos \frac{x}{2} = \left( \cos \frac{x}{2} + \sin \frac{x}{2} \right)^2 \]
This means we can rewrite the integral as:
\[ I = \int \sqrt{\left( \cos \frac{x}{2} + \sin \frac{x}{2} \right)^2} \, dx \]
Since the square root and square cancel each other, we have:
\[ I = \int \left( \cos \frac{x}{2} + \sin \frac{x}{2} \right) \, dx \]

### Step 2: Integrate the simplified expression
Now we can integrate the expression:
\[ I = \int \left( \cos \frac{x}{2} + \sin \frac{x}{2} \right) \, dx \]
We can split this into two separate integrals:
\[ I = \int \cos \frac{x}{2} \, dx + \int \sin \frac{x}{2} \, dx \]

### Step 3: Solve each integral
For the first integral:
\[ \int \cos \frac{x}{2} \, dx = 2 \sin \frac{x}{2} + C_1 \]
For the second integral:
\[ \int \sin \frac{x}{2} \, dx = -2 \cos \frac{x}{2} + C_2 \]

### Step 4: Combine the results
Combining both results, we have:
\[ I = 2 \sin \frac{x}{2} - 2 \cos \frac{x}{2} + C \]
where \( C = C_1 + C_2 \) is the constant of integration.

### Final Answer
Thus, the final result for the integral is:
\[ I = 2 \sin \frac{x}{2} - 2 \cos \frac{x}{2} + C \]