Question

How do you find all possible antiderivatives of a function?

Answer 1

Hello !

If `f` is function definite on a INTERVAL `I` (it's very important), and if `F` is a particular antiderivative of `f`, then all other antiderivative of `f` are

`F+c`

where `c` is a real constant.

Proof. If `G` is another antiderivative, then `G' = f = F'`, so `(G-F)' = f-f=0`. Because `I` is an interval, you can say that `G-F` is constant (so called `c`). Conclusion, `G=F+c`.

Remark. If `I` is not an interval, like `RR^\star`, it's false because you can have `f'=0` with `f` not constant.

Example : `f (x) = \frac{x}{|x|}` definite on `RR^\star` :
- if `x>0`, then `f(x) = 1`
- if `x<0`, then `f(x) = -` # - for all #x\in RR^\star`,`f'(x) = 0`, but`f`is not constant on`RR^\star#.