# Notation for Higher Antiderivatives?

Higher derivative are blessed with many notations.

For example $$f',f'',...$$ or $$\frac {dy}{dx}, \frac {d^2 y}{dx^2},...$$ I have not seen any notations for higher anti-derivatives.

For example, the higher anti-derivatives of $$f(x)=2x+5$$ are $$x^2+5x+c_1, \frac {x^3}{3} +\frac {5}{2} x^2 + c_1x +c_2,.....$$

Are there notations for higher anti-derivatives?

• I find usually if this is coming up, the boundary conditions are somehow built into the problem, so notations like $\int_0^x \int_0^{x'} f(x'') dx'' dx'$ become suitable. – Ian Dec 26 '18 at 18:21
• @Ian he said antiderivatives not integrals. – Ben W Dec 26 '18 at 18:22
• Interesting question. For the first few antiderivatives we just repeat the $\int$ symbol. In fact, latex allows us to write conveniently up to four of them using the \iiiint command, like so: $\iiiint$. At a certain point, of course, this becomes impractical. You could write it this way: $\underbrace{\int\int\cdots\int}_{n\text{ times}}$. But I wonder if there is something more compact. – Ben W Dec 26 '18 at 18:26
• @BenW Your last example becomes prettier if you still use the multiple integrals command: $\iint\cdots\int$ versus $\int\int\cdots\int$. – Arthur Dec 26 '18 at 19:20
• You may like one of the conventions in en.wikipedia.org/wiki/Fractional_calculus#Fractional_integrals – J.G. Dec 26 '18 at 19:35

Using Lagrange's notation as detailed in the link, $$\underbrace{\iint\cdots\int}_{n\text{ times}}=f^{(-n)}(x)$$ Using Newton's notation as detailed in the link, $$\underbrace{\iint\cdots\int}_{n\text{ times}}=\overset{n}{\overset{'}{y}}$$ although it "did not become widespread because of printing difficulties."