The search for antiderivative of a given function on an interval, that is, the search for a function \(F(x)\) for a given function \(f(x)\) on an interval \(I\) with the property that \(F'(x)=f(x)\), is called finding the primitive or integrating. The following equality is valid for this primitive function \(F\) and any \(a\) and \(x\) in \(I\) \[F(x)=F(a)+ \int_a^x f(\xi)\,\dd \xi.\] Every antiderivative (also called primitive) of \(f(x)\) on \(I\) can be written this way. One often denotes such a primitive function of \(f(x)\) without lower and upper limits in the format \[\int f(x)\,\dd x\] and then one callls it an indefinite integral. The indeterminacy lies in the fact that antiderivative of \(f(x)\) is only uniquely determined up to a constant of integration. A simple example of an indefinite integral on \(I=(-\infty,\infty)\) is \[\int (2x+3)\,\dd x = x^2+3x+c\] where the constant of integration is denoted with the letter \(c\). Many mathematical software programs such as the computer algebra systems Maple and Mathematica do not mention the constant of integration and provide only one primitive function.
Integration as the reverse of differentiation (34:14)