`intsqrt(1+sinx)dx`

Explanation

To solve the integral \(\int \sqrt{1 + \sin x} \, dx\), we can use trigonometric identities and substitution. Here’s a step-by-step solution:

### Step 1: Rewrite the integrand
We start by using the identity for \(1 + \sin x\):
\[ 1 + \sin x = \left(\sin^2 \frac{x}{2} + \cos^2 \frac{x}{2}\right) + 2 \sin \frac{x}{2} \cos \frac{x}{2} = \left(\sin \frac{x}{2} + \cos \frac{x}{2}\right)^2 \]
Thus, we can rewrite the integral:
\[ \sqrt{1 + \sin x} = \sqrt{\left(\sin \frac{x}{2} + \cos \frac{x}{2}\right)^2} = \sin \frac{x}{2} + \cos \frac{x}{2} \]

### Step 2: Set up the integral
Now, we can rewrite the integral:
\[ \int \sqrt{1 + \sin x} \, dx = \int \left(\sin \frac{x}{2} + \cos \frac{x}{2}\right) \, dx \]

### Step 3: Use substitution
Let \(t = \frac{x}{2}\), then \(dx = 2 \, dt\). Substituting this into the integral gives:
\[ \int \left(\sin t + \cos t\right) \cdot 2 \, dt = 2 \int \left(\sin t + \cos t\right) \, dt \]

### Step 4: Integrate
Now we can integrate:
\[ 2 \int \left(\sin t + \cos t\right) \, dt = 2 \left(-\cos t + \sin t\right) + C \]

### Step 5: Substitute back
Now we substitute back \(t = \frac{x}{2}\):
\[ 2 \left(-\cos \frac{x}{2} + \sin \frac{x}{2}\right) + C = 2\sin \frac{x}{2} - 2\cos \frac{x}{2} + C \]

### Final Result
Thus, the final result for the integral is:
\[ \int \sqrt{1 + \sin x} \, dx = 2\sin \frac{x}{2} - 2\cos \frac{x}{2} + C \]