Integral Types 2B1E99

1. **Stating the problem:** We want to understand the difference between an indefinite integral and a definite integral (often called a "normal integral"). 2. **Definition of indefinite integral:** The indefinite integral of a function $f(x)$ is the family of all antiderivatives of $f(x)$, written as $$\int f(x)\,dx = F(x) + C$$ where $F'(x) = f(x)$ and $C$ is an arbitrary constant called the constant of integration. 3. **Definition of definite integral:** The definite integral of $f(x)$ from $a$ to $b$ is the signed area under the curve $f(x)$ between $x=a$ and $x=b$, written as $$\int_a^b f(x)\,dx$$ and it evaluates to a number. 4. **Key differences:** - The indefinite integral represents a family of functions (antiderivatives) plus a constant. - The definite integral represents a number (area or accumulation) between two limits. 5. **Example:** - Indefinite integral: $$\int 2x\,dx = x^2 + C$$ - Definite integral: $$\int_0^2 2x\,dx = \left[x^2\right]_0^2 = 2^2 - 0^2 = 4$$ 6. **Summary:** - Indefinite integral: no limits, results in a function plus constant. - Definite integral: with limits, results in a number. This explains the difference between indefinite and definite (normal) integrals clearly.