Find the intervals in which `f(x)=s in\ x(1+cosx),\ \ 0

Explanation

To determine the intervals in which the function \( f(x) = \sin x (1 + \cos x) \) is increasing or decreasing on the interval \( 0 < x < \frac{\pi}{2} \), we will follow these steps:

### Step 1: Find the derivative \( f'(x) \)

To find the intervals of increase and decrease, we first need to compute the derivative of the function \( f(x) \).

Using the product rule:
\[
...