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I've seen some problems where the OP writes integrals in this form

$$\int {dt} f\left( t \right)$$

or for double integrals

$$\int {dx} \int {dtf\left( {t,x} \right)} $$

Do they represent another kind of integrals, or is it just notation?

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Just notation, as far as I know. It shows the variable of integration and avoids using brackets. I was taught that $dt$ may be written anywhere and may be treated (when it comes to notation) as just another factor. – savick01 Feb 23 '12 at 21:03
up vote 6 down vote accepted

This is just notation. In general, $$\int f(t) dt = \int dt f(t)$$

In fact you can move the $dt$ term anywhere you want--as long as it remains within its corresponding integral. So $$\int dx \int dt f(t, x) = \int \left( \int f(t, x) dt \right) dx = \int \int f(t, x) dt dx.$$

However, $$\int dx\,dt \color{red}{\int f(t, x)} \neq \int \int f(t, x) \, dt \, dx$$

because on the LHS, the right integral in the red has no $d$ term and thus is nonsense.

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