1. **State the problem:** We need to evaluate the integral $$\int 3a e^{6a} \, da$$. 2. **Formula and method:** This is an integration by parts problem. Recall the formula: $$\int u \, dv = uv - \int v \, du$$ We choose: - $u = 3a$ so that $du = 3 \, da$ - $dv = e^{6a} \, da$ so that $v = \frac{1}{6} e^{6a}$ (since $\int e^{kx} dx = \frac{1}{k} e^{kx}$) 3. **Apply integration by parts:** $$\int 3a e^{6a} \, da = 3a \cdot \frac{1}{6} e^{6a} - \int \frac{1}{6} e^{6a} \cdot 3 \, da$$ 4. **Simplify the expression:** $$= \frac{3a}{6} e^{6a} - \frac{3}{6} \int e^{6a} \, da$$ 5. **Cancel common factors:** $$= \frac{\cancel{3}a}{\cancel{6}} e^{6a} - \frac{\cancel{3}}{\cancel{6}} \int e^{6a} \, da = \frac{a}{2} e^{6a} - \frac{1}{2} \int e^{6a} \, da$$ 6. **Integrate remaining integral:** $$\int e^{6a} \, da = \frac{1}{6} e^{6a}$$ 7. **Substitute back:** $$= \frac{a}{2} e^{6a} - \frac{1}{2} \cdot \frac{1}{6} e^{6a} + C = \frac{a}{2} e^{6a} - \frac{1}{12} e^{6a} + C$$ 8. **Final answer:** $$\int 3a e^{6a} \, da = e^{6a} \left( \frac{a}{2} - \frac{1}{12} \right) + C$$