Find the maximum value of `4sin^2x+3cos^2x+sin(x/2)+cos(x/2)dot`

Explanation

To find the maximum value of the expression \( I = 4\sin^2 x + 3\cos^2 x + \sin\left(\frac{x}{2}\right) + \cos\left(\frac{x}{2}\right) \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ I = 4\sin^2 x + 3\cos^2 x + \sin\left(\frac{x}{2}\right) + \cos\left(\frac{x}{2}\right) \]
We know that \(\cos^2 x = 1 - \sin^2 x\). Therefore, we can rewrite \(I\) as:
...