1. Statement of the problem: Compute the integral $\int (12x^6+7x^5+2)\,dx$. 2. Reasoning: Use linearity to integrate term-by-term and the power rule $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$ for $n\neq-1$. 3. Integrate each term separately. Apply to the first term: $\int 12x^6\,dx=12\int x^6\,dx=12\cdot\frac{x^7}{7}=\frac{12x^7}{7}$. Apply to the second term: $\int 7x^5\,dx=7\int x^5\,dx=7\cdot\frac{x^6}{6}=\frac{7x^6}{6}$. Apply to the constant term: $\int 2\,dx=2x$. 4. Combine the results and add the constant of integration. $$\int (12x^6+7x^5+2)\,dx=\frac{12x^7}{7}+\frac{7x^6}{6}+2x+C$$