Integral Basics 16B486

1. **Stating the problem:** We want to understand the concept of an integral in mathematics at level 5 TVET school. 2. **What is an integral?** An integral is a fundamental concept in calculus that represents the area under a curve or the accumulation of quantities. 3. **Types of integrals:** There are two main types: definite integrals and indefinite integrals. 4. **Indefinite integral:** It represents a family of functions and is written as $$\int f(x)\,dx = F(x) + C$$ where $F'(x) = f(x)$ and $C$ is the constant of integration. 5. **Definite integral:** It calculates the area under the curve $f(x)$ from $a$ to $b$ and is written as $$\int_a^b f(x)\,dx = F(b) - F(a)$$ where $F$ is an antiderivative of $f$. 6. **Important rules:** - Linearity: $$\int (af(x) + bg(x))\,dx = a\int f(x)\,dx + b\int g(x)\,dx$$ - Power rule: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$ 7. **Example:** Find $$\int (3x^2 + 2x)\,dx$$ 8. **Solution:** $$\int 3x^2\,dx + \int 2x\,dx = 3 \cdot \frac{x^{3}}{3} + 2 \cdot \frac{x^{2}}{2} + C = x^{3} + x^{2} + C$$ 9. **Explanation:** We integrated each term separately using the power rule and added the constant $C$. This is the basic understanding of integrals at level 5 TVET school.