`int(cos((x)/(2))-sin((x)/(2)))^2dx=`

Options

A. `x+cosx+c`
B. `2cos^2((x)/(2))+c`
C. `(1)/(3)(cos((x)/(2))-(x)/(2))^3+c`
D. `x-cosx+c`

Explanation

To solve the integral \( \int \left( \cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right) \right)^2 \, dx \), we will follow these steps:

### Step 1: Expand the Square
We start by expanding the expression inside the integral:
\[ \left( \cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right) \right)^2 = \cos^2\left(\frac{x}{2}\right) - 2\cos\left(\frac{x}{2}\right)\sin\left(\frac{x}{2}\right) + \sin^2\left(\frac{x}{2}\right) \]

### Step 2: Use Trigonometric Identity
Using the identity \( \cos^2 A + \sin^2 A = 1 \):
\[ \cos^2\left(\frac{x}{2}\right) + \sin^2\left(\frac{x}{2}\right) = 1 \]
Thus, we can rewrite the integral as:
\[ \int \left( 1 - 2\cos\left(\frac{x}{2}\right)\sin\left(\frac{x}{2}\right) \right) \, dx \]

### Step 3: Simplify the Expression
Recognizing that \( 2\cos A \sin A = \sin(2A) \):
\[ -2\cos\left(\frac{x}{2}\right)\sin\left(\frac{x}{2}\right) = -\sin(x) \]
So, the integral simplifies to:
\[ \int \left( 1 - \sin(x) \right) \, dx \]

### Step 4: Integrate
Now we can integrate term by term:
\[ \int 1 \, dx - \int \sin(x) \, dx = x + \cos(x) + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final result of the integral is:
\[ \int \left( \cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right) \right)^2 \, dx = x + \cos(x) + C \]