Integrate the following :
`intcos^(2) theta d theta`

Explanation

To solve the integral \( \int \cos^2 \theta \, d\theta \), we can use a trigonometric identity to simplify the integrand. Here are the steps:

### Step 1: Use the Trigonometric Identity
We know that:
\[ \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \]
This identity allows us to express \( \cos^2 \theta \) in a form that is easier to integrate.

### Step 2: Substitute the Identity into the Integral
Substituting the identity into the integral gives:
\[ \int \cos^2 \theta \, d\theta = \int \frac{1 + \cos 2\theta}{2} \, d\theta \]

### Step 3: Factor Out the Constant
We can factor out the \( \frac{1}{2} \):
\[ = \frac{1}{2} \int (1 + \cos 2\theta) \, d\theta \]

### Step 4: Split the Integral
Now we can split the integral into two parts:
\[ = \frac{1}{2} \left( \int 1 \, d\theta + \int \cos 2\theta \, d\theta \right) \]

### Step 5: Integrate Each Part
Now we integrate each part:
1. The integral of \( 1 \) with respect to \( \theta \) is \( \theta \).
2. The integral of \( \cos 2\theta \) is \( \frac{\sin 2\theta}{2} \) (using the substitution \( u = 2\theta \)).

Putting it all together:
\[ = \frac{1}{2} \left( \theta + \frac{\sin 2\theta}{2} \right) + C \]

### Step 6: Simplify the Expression
Now, we simplify the expression:
\[ = \frac{\theta}{2} + \frac{\sin 2\theta}{4} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \cos^2 \theta \, d\theta = \frac{\theta}{2} + \frac{\sin 2\theta}{4} + C \]