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

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`

Answer

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

### Step 1: Expand the integrand
We start by expanding the expression \( \left( \cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right) \right)^2 \).

Using the identity \( (A - B)^2 = A^2 - 2AB + B^2 \), we have:
\[ \left( \cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right) \right)^2 = \cos^2\left(\frac{x}{2}\right) - 2 \cos\left(\frac{x}{2}\right) \sin\left(\frac{x}{2}\right) + \sin^2\left(\frac{x}{2}\right) \]

### Step 2: Simplify the expression
Now, we can simplify the expression using the Pythagorean identity:
\[ \cos^2\left(\frac{x}{2}\right) + \sin^2\left(\frac{x}{2}\right) = 1 \]
Thus, the expression simplifies to:
\[ 1 - 2 \cos\left(\frac{x}{2}\right) \sin\left(\frac{x}{2}\right) \]

### Step 3: Use the double angle identity
We can use the double angle identity for sine:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
In our case, \( \theta = \frac{x}{2} \), so:
\[ 2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) = \sin(x) \]
Thus, we rewrite the integrand:
\[ 1 - \sin(x) \]

### Step 4: Integrate the simplified expression
Now we can integrate:
\[ \int (1 - \sin(x)) \, dx \]

This can be split into two separate integrals:
\[ \int 1 \, dx - \int \sin(x) \, dx \]

Calculating these integrals:
1. \( \int 1 \, dx = x \)
2. \( \int \sin(x) \, dx = -\cos(x) \)

Putting it all together:
\[ \int (1 - \sin(x)) \, dx = x + \cos(x) + C \]

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

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