1. The problem is to find the integral of $\sin^2 x$ with respect to $x$. 2. We use the trigonometric identity to simplify the integrand: $$\sin^2 x = \frac{1 - \cos(2x)}{2}$$ 3. Substitute this into the integral: $$\int \sin^2 x \, dx = \int \frac{1 - \cos(2x)}{2} \, dx = \frac{1}{2} \int (1 - \cos(2x)) \, dx$$ 4. Split the integral: $$\frac{1}{2} \int 1 \, dx - \frac{1}{2} \int \cos(2x) \, dx$$ 5. Integrate each term: - $$\int 1 \, dx = x$$ - $$\int \cos(2x) \, dx = \frac{\sin(2x)}{2}$$ (using substitution or standard integral rules) 6. Substitute back: $$\frac{1}{2} x - \frac{1}{2} \cdot \frac{\sin(2x)}{2} + C = \frac{x}{2} - \frac{\sin(2x)}{4} + C$$ 7. Therefore, the integral is $$\int \sin^2 x \, dx = \frac{x}{2} - \frac{\sin(2x)}{4} + C$$ where $C$ is the constant of integration.