7 Appllied of Derivatives
7.1 Indefinite Integrals
Indefinite integrals, or antiderivatives, undo the process of differentiation, enabling us to recover the original function from its derivative. They represent accumulated quantities such as displacement, total growth, or total charge. The Table 7.1 below summarizes key concepts, descriptions, and example applications see also [1], [2].
| KeyConcept | Description | ExampleApplication |
|---|---|---|
| Definition | \(F'(x) = f(x)\); \(\int f(x) dx = F(x) + C\) | Displacement: \(s(t) = \int v(t) dt\); e.g., \(v(t)=3t^2 \implies s(t)=t^3+C\) |
| Power Rule | \(\int x^n dx = \frac{x^{n+1}}{n+1}+C, n\neq -1\) | Integration of \(x^2\) gives \(\frac{x^3}{3}+C\) |
| Constant Multiple | \(\int c f(x) dx = c \int f(x) dx\) | Multiply constant with integral |
| Sum Rule | \(\int [f(x)+g(x)] dx = \int f(x) dx + \int g(x) dx\) | \(\int (x^2+2x) dx = \frac{x^3}{3}+x^2+C\) |
| Total Accumulation | Accumulation of quantity over time | Revenue: \(\int R(t) dt\) |
References
[1]
Stewart, J., Calculus: Early transcendentals, Cengage Learning, 2016
[2]
Apostol, T. M., Calculus, volume i: One-variable calculus with an introduction to linear algebra, Wiley, 1967