# Thread: Notation for the nth Integral

1. ## Notation for the nth Integral

If we want to represent the nth derivative of $\displaystyle (ax+b)^m$, then out of many we can write:

$\displaystyle \displaystyle f^{(n)}(x) = a^n(ax+b)^{m-n}\prod_{n=0}^{n-1}(m-n)$ (Lagrange's notation).

$\displaystyle \displaystyle \frac{d^ny}{dx^n} = a^n(ax+b)^{m-n}\prod_{n=0}^{n-1}(m-n)$ (Leibniz's notation).

But what notation do we use for representing the reverse — that is, for the nth integral of the RHS?

2. Is there in anything wrong/ambiguous about using $\displaystyle \int f(x)\;{dx_{n}}$ for the nth integral of $\displaystyle f(x)$?

[Got that idea from wikipedia article that says the multiple integration of a function in $\displaystyle n$ variables,
$\displaystyle f(x_1, x_2, ..., x_n)$, over a domain D is represented by $\displaystyle \int \cdots \int_\mathbf{D}\;f(x_1,x_2,\ldots,x_n) \;\mathbf{d}x_1 \!\cdots\mathbf{d}x_n$]

3. You will also sometimes see $\displaystyle D^n(f)$ for the nth derivative and $\displaystyle D^{-n}(f)$ for the nth anti-derivative. Of course, $\displaystyle D^0(f)= f$. If you use $\displaystyle f^{(n)}(x)$ for the nth derivative, you could use $\displaystyle f^{(-n)}(x)$ for the nth anti-derivative.

TheCoffeeMachine, wouldn't the form you give be likely to be confused with the integral with respect to the nth coordinate variable?

4. Originally Posted by TheCoffeeMachine
Is there in anything wrong/ambiguous about using $\displaystyle \int f(x)\;{dx_{n}}$ for the nth integral of $\displaystyle f(x)$?

[Got that idea from wikipedia article that says the multiple integration of a function in $\displaystyle n$ variables,
$\displaystyle f(x_1, x_2, ..., x_n)$, over a domain D is represented by $\displaystyle \int \cdots \int_\mathbf{D}\;f(x_1,x_2,\ldots,x_n) \;\mathbf{d}x_1 \!\cdots\mathbf{d}x_n$]
Yes, because what you have written refers to integrating a vector function with respect to the nth component.