1. **Stating the problem:** Integration is a fundamental concept in calculus used to find areas under curves, accumulated quantities, and antiderivatives. 2. **Formula and rules:** The integral of a function $f(x)$ with respect to $x$ is denoted as $$\int f(x)\,dx$$ and represents the family of all antiderivatives of $f(x)$ plus a constant $C$. 3. **Basic integration rules:** - Power rule: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$ - Constant multiple rule: $$\int a f(x)\,dx = a \int f(x)\,dx$$ - Sum rule: $$\int (f(x) + g(x))\,dx = \int f(x)\,dx + \int g(x)\,dx$$ 4. **Example:** Find $$\int (3x^2 + 2x + 1)\,dx$$ 5. **Step-by-step solution:** - Apply sum rule: $$\int 3x^2\,dx + \int 2x\,dx + \int 1\,dx$$ - Apply constant multiple rule: $$3 \int x^2\,dx + 2 \int x\,dx + \int 1\,dx$$ - Use power rule: $$3 \cdot \frac{x^{2+1}}{2+1} + 2 \cdot \frac{x^{1+1}}{1+1} + x + C$$ - Simplify: $$3 \cdot \frac{x^3}{3} + 2 \cdot \frac{x^2}{2} + x + C$$ - Cancel common factors: $$\cancel{3} \cdot \frac{x^3}{\cancel{3}} + \cancel{2} \cdot \frac{x^2}{\cancel{2}} + x + C$$ - Final answer: $$x^3 + x^2 + x + C$$ 6. **Summary:** Integration reverses differentiation and helps find areas and accumulated values. Remember to add the constant of integration $C$ because derivatives of constants are zero.