`intcos^(2) x dx`

Answer

To solve the integral \( \int \cos^2 x \, dx \), we can use the trigonometric identity for cosine. Here’s the step-by-step solution:

### Step 1: Use the identity for \( \cos^2 x \)
We know that:
\[ \cos^2 x = \frac{1 + \cos(2x)}{2} \]
This identity allows us to rewrite the integral.

### Step 2: Substitute the identity into the integral
Now we can substitute this identity into our integral:
\[ \int \cos^2 x \, dx = \int \frac{1 + \cos(2x)}{2} \, dx \]

### Step 3: Factor out the constant
We can factor out the \( \frac{1}{2} \):
\[ = \frac{1}{2} \int (1 + \cos(2x)) \, dx \]

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

### Step 5: Integrate each part
1. The integral of \( 1 \) is simply \( x \).
2. The integral of \( \cos(2x) \) can be solved using the substitution method or known results:
\[ \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) \]

Putting these results together:
\[ = \frac{1}{2} \left( x + \frac{1}{2} \sin(2x) \right) \]

### Step 6: Simplify the expression
Now, we can simplify the expression:
\[ = \frac{x}{2} + \frac{1}{4} \sin(2x) + C \]
where \( C \) is the constant of integration.

### Final Answer:
Thus, the final result is:
\[ \int \cos^2 x \, dx = \frac{x}{2} + \frac{1}{4} \sin(2x) + C \]