Defining an mathematical operator

I’m trying to learn mathematica to use in my thesis. I have basic problems with definitions, calculations etc. Let me ask a question.

I want to define anti derivative operator and derivative operator, then a new operator with composition of these operators. Compositions must be identity, so if I give a function into new operator it will turn out the function what I gave it.

To explain better, I will try to write these in mathematical sense;

Here $D^m$ denotes standard derivative operator $m$ times and anti-derivative operator $I_m$ defined with

$
qquad begin{cases}
f(x), & m=0
int_0^{x} dfrac{(x-y)^{m-1}}{(m-1)!}f(u)du, & mgeq 1
end{cases}
$

Now, if we composite these functions, $[D^mcirc I_m](f)(x)$ these will give us directly $f(x)$. I want to define these operators in Mathematica and see composition of operators.

Actually these problem is not my main problem, but the answers will shows me what I need, then I will try to learn myself again.