The antiderivative (sometimes also called the indefinite integral) of a function (usually denoted with the uppercase of the function, in this case ) is a differentiable function who's derivative is the original function .

The operation of taking an antiderivative is usually denoted with the same symbol as an integral but without the upper and lower bounds.

For example:

The Constant of Integration

Because of constants being "lost" when a function's derivative is taken, when antiderivatives are calculated they need to have an arbitrary constant added to them (usually denoted with a ).

For example when the derivative of the function is taken the result is . There is no way to determine the constant of the original function without knowing the original function therefore when the antiderivative of is taken a term is added making the antiderivative of equal .

Finding the Constant

The constant can be found when you're provided with some point on the anti-derivatives graph (usually at ). When this point is at it is called the "initial condition".

Example

Without an Initial Condition

Given the function find the anti-derivative of .

With an Initial Condition

Given the function find the anti-derivative of with the initial condition that .

Calculating

Antiderivatives can be calculated through knowledge of the different differentiation rulesalternatively antiderivatives can be taken for more complex functions using different antiderivative techniques. Additionally certain antiderivatives can be memorized to speed antiderivatives up.