Integral Example 76F2E9

1. **Problem Statement:** Calculate the integral $$\int (3x^2 + 2x + 1) \, dx$$. 2. **Formula Used:** The integral of a polynomial term $$ax^n$$ is given by $$\int ax^n \, dx = \frac{a}{n+1}x^{n+1} + C$$, where $$C$$ is the constant of integration. 3. **Step-by-step Solution:** - Break the integral into separate terms: $$\int (3x^2 + 2x + 1) \, dx = \int 3x^2 \, dx + \int 2x \, dx + \int 1 \, dx$$ - Integrate each term: $$\int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3}$$ - Simplify by canceling common factors: $$3 \cdot \frac{x^3}{\cancel{3}} = \cancel{3} \cdot \frac{x^3}{3} = x^3$$ - Next term: $$\int 2x \, dx = 2 \int x \, dx = 2 \cdot \frac{x^{1+1}}{1+1} = 2 \cdot \frac{x^2}{2}$$ - Simplify by canceling common factors: $$2 \cdot \frac{x^2}{\cancel{2}} = \cancel{2} \cdot \frac{x^2}{2} = x^2$$ - Last term: $$\int 1 \, dx = x$$ 4. **Combine all results:** $$x^3 + x^2 + x + C$$ 5. **Final Answer:** $$\int (3x^2 + 2x + 1) \, dx = x^3 + x^2 + x + C$$ This shows how to integrate a polynomial by applying the power rule term-by-term and simplifying.