1. The problem is to find the integral of $\cos^2 \theta$ with respect to $\theta$. 2. Use the trigonometric identity to simplify the integrand: $$\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}$$ 3. Substitute this into the integral: $$\int \cos^2 \theta \, d\theta = \int \frac{1 + \cos(2\theta)}{2} \, d\theta = \frac{1}{2} \int (1 + \cos(2\theta)) \, d\theta$$ 4. Split the integral: $$\frac{1}{2} \int 1 \, d\theta + \frac{1}{2} \int \cos(2\theta) \, d\theta$$ 5. Integrate each term: - $\int 1 \, d\theta = \theta$ - $\int \cos(2\theta) \, d\theta = \frac{\sin(2\theta)}{2}$ (using substitution) 6. Combine the results: $$\frac{1}{2} \theta + \frac{1}{2} \cdot \frac{\sin(2\theta)}{2} + C = \frac{\theta}{2} + \frac{\sin(2\theta)}{4} + C$$ 7. Therefore, the integral of $\cos^2 \theta$ is: $$\int \cos^2 \theta \, d\theta = \frac{\theta}{2} + \frac{\sin(2\theta)}{4} + C$$