1. The problem asks us to find the indefinite integral of the function $12x^6 + 7x^5 + 2$ with respect to $x$. 2. We apply the power rule for integration to each term separately. Recall the power rule: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C,$$ where $C$ is the constant of integration. 3. Integrate each term: - For $12x^6$, we have $$\int 12x^6 \, dx = 12 \cdot \frac{x^{6+1}}{6+1} = 12 \cdot \frac{x^7}{7} = \frac{12}{7} x^7.$$ - For $7x^5$, we have $$\int 7x^5 \, dx = 7 \cdot \frac{x^{5+1}}{5+1} = 7 \cdot \frac{x^6}{6} = \frac{7}{6} x^6.$$ - For the constant $2$, recall that $$\int a \, dx = ax + C,$$ so $$\int 2 \, dx = 2x.$$ 4. Combine all integrated terms and add the constant of integration $C$: $$\int (12x^6 + 7x^5 + 2) \, dx = \frac{12}{7} x^7 + \frac{7}{6} x^6 + 2x + C.$$