Antiderivatives

The antiderivative of a function \(f\) is any differentiable function \(F\) such that \(F'=f\). However, the antiderivative, if it exists, is not unique. In fact, if \(f\) has one antiderivative, then it has infinitely many, for if \(F'=f\), then \((F+c)'=f\) for every constant \(c\), since the constant disappears upon taking the derivative.

The indefinite integral \(\int f(x) \mathrm{d}x\) can be considered as a function mapping functions to their antiderivatives. To be mathematicaly precise then, if \(F\) is an antiderivative of \(f\), we have to write

\[\int f(x) \mathrm{d}x = F +c \]

where \(c\) can be any constant. When asked to find the antiderivatives of a function, one always has to write it this way to be precise. Otherwise, one only finds a particular solution; remove the \(c\) and you will remove infinitely many correct answers!

When solving definite integrals by using the Fundamental Theorem of Calculus and antiderivatives however, one may disregard the constant as the following obviously holds for any constant:

\[(F(b)+c)-(F(a)+c)=F(b)-F(a)+c-c=F(b)-F(a).\]

2013-06-15, mathiaar