1. We are asked to find the integral of $\sin^2(x) \cos(x) \, dx$. 2. The integral is $\int \sin^2(x) \cos(x) \, dx$. 3. Notice that the integrand contains $\sin^2(x)$ and $\cos(x)$, and the derivative of $\sin(x)$ is $\cos(x)$, so we can use substitution. 4. Let $u = \sin(x)$, then $du = \cos(x) \, dx$. 5. Substitute into the integral: $\int u^2 \, du$. 6. Integrate: $\int u^2 \, du = \frac{u^3}{3} + C$. 7. Substitute back $u = \sin(x)$: $$\frac{\sin^3(x)}{3} + C$$. Final answer: $$\int \sin^2(x) \cos(x) \, dx = \frac{\sin^3(x)}{3} + C$$