Antiderivative

An antiderivative of a function on an interval is a function whose derivative equals that function throughout the interval. Antiderivatives differ by constants on connected intervals.

Derivative: The derivative of a function gives its instantaneous rate of change with respect to its input. An antiderivative is defined by having this function as its derivative.
Power rule: The power rule gives the derivative of a power function and, in reverse, a basic rule for integrating powers. Reversing its derivative formula produces the power rule for antiderivatives.
Definite integral: A definite integral gives the signed accumulation of a function over a specified interval. An antiderivative evaluates it through the fundamental theorem of calculus.
Numerical integration: Numerical integration approximates definite integrals using finite calculations from function values. It can estimate accumulated area without finding an explicit antiderivative.
Indefinite integral: The indefinite integral represents the family of antiderivatives of a function, usually written with an arbitrary constant. It is the standard notation for collecting all antiderivatives.
Exponential function: An exponential function has a constant base raised to a variable exponent; the natural exponential is its own derivative. Its natural-base form is an antiderivative of itself.
Initial condition: An initial condition specifies the value of an unknown function at a particular input. It fixes the additive constant when recovering a function from its derivative.
Non-elementary antiderivative: A non-elementary antiderivative cannot be expressed using a finite combination of elementary functions. Some functions have antiderivatives even when no elementary formula exists.
Constant of integration: The constant of integration represents the freedom to add any constant to an antiderivative without changing its derivative. It accounts for why a function’s antiderivatives form a family rather than a single function.
Natural logarithm: The natural logarithm is the inverse of the natural exponential function and has derivative 1/x for x greater than zero. It supplies a standard antiderivative for reciprocal functions on positive intervals.

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