1. **State the problem:** We need to evaluate the integral $$\int 12 \cos^2 x \sin x \, dx$$. 2. **Recall the formula and substitution:** Notice the integral involves powers of cosine and sine. A useful substitution is to let $$u = \cos x$$, then $$du = -\sin x \, dx$$ or equivalently $$-du = \sin x \, dx$$. 3. **Rewrite the integral using substitution:** $$\int 12 \cos^2 x \sin x \, dx = 12 \int \cos^2 x \sin x \, dx$$ Substitute $$u = \cos x$$ and $$\sin x \, dx = -du$$: $$= 12 \int u^2 (-du) = -12 \int u^2 \, du$$ 4. **Integrate with respect to $$u$$:** $$-12 \int u^2 \, du = -12 \cdot \frac{u^3}{3} + C = -4 u^3 + C$$ 5. **Back-substitute $$u = \cos x$$:** $$-4 \cos^3 x + C$$ **Final answer:** $$\int 12 \cos^2 x \sin x \, dx = -4 \cos^3 x + C$$