What is the correct notation for integrating F(x)?

[majin_andrew]

Hi, this isn't a homework question, I'm just curious about this.

I am wondering what the correct notation is for the integral of F(x).

For example,
integral of f''(x) = f'(x) + c
integral of f'(x) = f(x) + d
integral of f(x) = F(x) + e
integral of F(x) = ??

I feel silly for not knowing this. Is there a common notation that I am not aware of, or is it simply a case of letting F(x) = g''(x) (or whatever) and carrying on from there?

Thanks!
Andrew

 

[Dick]

There isn't any standard notation for that. Even integral f(x)=F(x)+C isn't really standard. It's common to write that when you are being introduced to antiderivatives, but in general if you write 'F(x)' you should never assume someone will automatically know it's the integral of f(x).

 

[majin_andrew]

Edit: Sorry had trouble with the equation editing

 

[majin_andrew]

Okay thanks for that Dick. So if I would like to write the second integral of f(x), is it the proper notation to write it as $$\int{\int{f(x)d^2 x^2}$$ ?

 

[Dick]

Okay thanks for that Dick. So if I would like to write the second integral of f(x), is it the proper notation to write it as $$\int{\int{f(x)d^2 x^2}$$ ?

Even the first integral should really be written $$F(x)=\int_a^x f(t)dt$$. The x dependence is really in the limit not in the dummy integration variable. $$\int f(x) dx$$ is really pretty casual. And you are REALLY stretching the definition of casual with that notation. Seriously, you very seldom need a 'second antiderivative' of f(x). That's probably why there is no good notation. If you do need it then just write 'pick g(x) to be a function such that g''(x)=f(x)'. Something like that.

 

[majin_andrew]

Ok, thanks for your help.

 

[slider142]

Although there are few cases where it is needed, the Jⁿ notation is used frequently in texts on integral equations and D^-n in texts on fractional calculus, each for repeated integration.