`intcos^(2) x dx`

Answer

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

### Step 1: Use the Trigonometric Identity
We know from trigonometric identities that:
\[ \cos 2x = 2 \cos^2 x - 1 \]
From this, we can express \( \cos^2 x \) in terms of \( \cos 2x \):
\[ \cos^2 x = \frac{\cos 2x + 1}{2} \]

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

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

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

### Step 5: Integrate Each Part
Now we can integrate each part separately:
- The integral of \( \cos 2x \) is:
\[ \int \cos 2x \, dx = \frac{\sin 2x}{2} \]
- The integral of \( 1 \) is:
\[ \int 1 \, dx = x \]

### Step 6: Combine the Results
Putting it all together, we have:
\[ \int \cos^2 x \, dx = \frac{1}{2} \left( \frac{\sin 2x}{2} + x \right) + C \]
This simplifies to:
\[ \int \cos^2 x \, dx = \frac{\sin 2x}{4} + \frac{x}{2} + C \]

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

---