Antideriválás

Antideriválás, also known as integration, is a fundamental concept in calculus. It is the inverse operation of differentiation. If a function $f(x)$ is the derivative of another function $F(x)$, then $F(x)$ is called an antiderivative or primitive of $f(x)$. The process of finding an antiderivative is called antideriválás.

When we find an antiderivative, there is an arbitrary constant of integration, usually denoted by $C$, that can be added to the result. This is because the derivative of a constant is always zero. Therefore, if $F(x)$ is an antiderivative of $f(x)$, then $F(x) + C$ is also an antiderivative of $f(x)$ for any constant $C$. The collection of all antiderivatives of a function is called its indefinite integral, denoted by $\int f(x) \, dx = F(x) + C$.

Antideriválás is crucial for solving many problems in mathematics, physics, engineering, and economics. For example, it is used to find the area under a curve, to solve differential equations, and to calculate work done by a variable force. The Fundamental Theorem of Calculus establishes a direct link between differentiation and integration, showing that definite integrals can be evaluated by finding an antiderivative. Common techniques for finding antiderivatives include substitution, integration by parts, and partial fraction decomposition.