The integral is one of the main operators in calculus, which is usually split into derivative calculus and integral calculus. Visually the integral of a function can be interpreted as the area under the curve between two points, where the area underneath the x-axis is considered negative.
The definite integral calculates the accumulated sum (area) under the curve of a function over a specific interval [a, b]. It can be denoted as ∫[a, b] f(x) dx. The definite integral results in a numerical value, which represents the net area between the function’s curve and the x-axis. This area is considered positive when the function lies above the x-axis and negative when it lies below.
The indefinite integral, also known as an antiderivative, represents the family of functions that have the given function as their derivative. It does not have specific limits and is denoted as . The result of an indefinite integral is a function, plus an arbitrary constant , since taking the derivative of a constant results in zero. The indefinite integral provides a general expression for calculating the area under the curve of the function over any interval.
The table below shows some common integrals. Each is an antiderivative of the function in the left column, so taking the derivative of the right column returns the original function. Every result includes the arbitrary constant .
| Function | Integral |
|---|---|
| Constant | |
| Line | |
| Power | |
| Reciprocal | |
| Sine | |
| Cosine | |
| Exponential | |
| Logarithm |
The power rule covers every exponent except . That one missing case is the reciprocal , which integrates to the natural logarithm rather than another power of .
Shown below are some common rules for integrating more complex functions. The sum and constant multiple rules mirror their counterparts for derivatives, while integration by parts and substitution reverse the product rule and the chain rule.
| Name | Rule |
|---|---|
| Sum Rule | |
| Constant Multiple | |
| Power Rule | |
| Integration by Parts | |
| Substitution |
The derivative is one of the main operators in calculus. It is used to find the rate of change of a function with respect to its input variable.
The limit operator describes the result of an expression as a variable approaches a value. The operator is used in calculus to formalize what mathematicians mean by approach.