The correct mathematical term for nth integral

Question

I'm looking for a correct English term for the following concept: $$ If(x)=\int_0^x f(t)dt \\ g(x) = I(If(x)) \\ h(x) = I^{n}f(x) $$

What should be the precise name of $h(x)$? So far I came up with:

• n-th order repeated integral
• n-th order integral
• nth order integral
• nth integral

Thank you for your help in advance

Answer 1

One could probably call it any of those terms, but the interesting thing is how far it simplifies. Thanks to Cauchy's repeated integral formula, we have

$$I^nf(x)=\frac1{(n-1)!}\int_0^x(x-t)^{n-1}f(t)\ dt$$

which is very helpful for deducing things like

$$\int_1^x\ln(t)\ dt=\int_1^x\int_1^t\frac1y\ dy\ dt=\frac1{1!}\int_1^x\frac{x-t}t\ dt=x\ln(x)-x+1$$

which is rather simple.

Answer 2

$f^{(-n)}(t)$ means nth antiderivative (indefinite integral) and $f^{(n)}(t)$ means nth derivative of $f(t)$

Write the notations as power of f written in round bracket. Positive power stands for derivative while negative power stands for antiderivative.

When n=0 , it would mean the function itself. I have seen these notations in many places.