I am trying to integrate the following triple integral, which has a heaviside function in the inner most integral:
$$ \frac{16}{c_{4}^{4}} \int_{0}^{c_{4}} c_{3}dc_{3} \int_{c_{3}}^{c_{4}} \frac{dc_{2}}{c_{2}} \int_{0}^{c_{2}}f(x)\left ( 1-H\left ( x-\left ( c_{4}-a \right ) \right ) \right )dx $$
where $f(x)=x$. I know $c_{2}>0$, $c_{3}>0$ , $c_{4}>0$, $x>0$ , $a>0 $, $c_{4}>a$
I get the right answer (which I know already) using symbolic integration and the heaviside function in Matlab, which is:
$$1-4\frac{a^{2}}{c_{4}^{2}}+4\frac{a^{3}}{c_{4}^{3}}-\frac{a^{4}}{c_{4}^{4}}$$
However, it is not clear to me how to do this manually?
Edit: When I simply try splitting the inner most integral into separate integrals, splitting the range, and one of them equaling zero for the value $ x<c_{4}-a$, I am still not getting the right answer...Does the Heaviside carry through the integrals somehow?
Many thanks in advance!