Antiderivative

Antidifferentiation (also called indefinite integration) is the process of finding a certain function in calculus. It is the opposite of differentiation. It is a way of processing a function to give another function (or class of functions) called an antiderivative. Antidifferentiation is like integration—but without limits. This is why it is called indefinite integration. When represented as single letters, antiderivatives often take the form of capital roman letters such as formula_1 and formula_2.
In general, an antiderivative is written in the form formula_3, where:
Simple antidifferentiation.
A function of the form formula_8 can be integrated (antidifferentiated) as follows:
This can be shown as:
formula_15 (also known as the power rule of integral)
When there are many terms, we can integrate the entire function by integrating its components one by one:
formula_16
Examples.
Changing fractions and roots into powers makes it easier:
Integrating a bracket ("chain rule").
To integrate a bracket like formula_22, a different method is needed. It is called the chain rule. It is like simple integration, but it only works if the formula_5 in the bracket is linear (has a power of 1), such as formula_5 or formula_25—but "not" formula_26 or formula_27.
For example, formula_28 can be determined in the following steps:
Examples.
formula_36
formula_37