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Many times/infinitely differentiable exists.

Does such multiple integration exist? can you take multiple integrals in one variable? (not double/triple integration/Fubini's theorem/vector calculus/multivariable calculus)

I want to get from acceleration to displacement, for example.

EDIT:

I've been told this is the correct notation: $$\int \int f(t) \,dt dt$$

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    Whats wrong with calling it 'Integrating twice'? Also, I guess you mean from acceleration to displacement. – 2017-02-23
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    Well, first of all it should be $dt$ by convention since we are working with time here. Secondly and more importantly, you have to have a differential for each integral. I've seen the notation $dt\;dt$ before, though perhaps $dt_1 dt_2$ would be more conventional – 2017-02-23
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    because when I said multiple integration, people think I'm saying double integral: $$\int \int f(x,y) \,dy dx$$ – 2017-02-23

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The accepted notation I've seen would simply be $$\underbrace{\int\cdots\int_0^x}_n f(x)\,\underbrace{dx\cdots dx}_n$$ See the Wolfram Mathworld page on Repeated Integration for example. In other words, you can simply write $dx\;dx$ if you want to.

Note that by Cauchy's Repeated Integration Formula we can write this as $$\frac{1}{(n-1)!}\int_0^x f(t)(x-t)^{n-1}dt$$

Another common notation can be found on Wikipedia's page on Cauchy's Repeated Integration Formula, namely $$ \int_0^x \int_0^{t_1} \cdots \int_0^{t_{n-1}} f(t) \, \mathrm{d}t_{n} \cdots \, \mathrm{d}t_2 \, \mathrm{d}t_1$$ Note that I have modified this slightly from Wikipedia

Here we can avoid using an underbrace when we have too many integrals to write out explicitly, because of the subscripts on the different differentials. Note that the outside integral as a limit of $x$ and so our function ends up being in terms of $x$ as it should

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    oh okay, so it's called a **repeated integration**. Thanks! – 2017-02-23
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    @user29418 No problem :) I also added the other notation I included in my comments above, where the differentials are numbered. This is done on the [Wikipedia page for Cauchy Repeated Integration](https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration) for example. – 2017-02-23