Sometimes, in order to prove an identity about the integral of a function we can apply multiple time integration by parts until we integrate an easy function.
For example it's a nice technique with polynomial. Let's say I want to calculate the following :
$$\int_{-1}^1 \left(\frac{1}{2^j j!}\right)^2 \left(\frac{d^j}{dx^j} (x^2-1)^j\right)^2 \mathrm{d}x$$
Then clearly I can do integration by part multiple times until I get an easy integral. So here by doing one integration by part the problem boils down to calculating :
\begin{align*} &\left(\frac{1}{2^jj!}\right)^2 \Bigg( \left[\left(\frac{d^{j-1}}{dx^{j-1}} (x^2-1)^j\right)\frac{d^j}{dx^j} (x^2-1)^j\right]_{-1}^1 \\ &\hspace{5em} -\int_{-1}^1 \left(\frac{d^{j-1}}{dx^{j-1}} (x^2-1)^j\right)\left(\frac{d^{j+1}}{dx^{j+1}} (x^2-1)^j\right) \mathrm{d}x\Bigg) \\ &= \left(\frac{1}{2^jj!}\right)^2 \int_{-1}^1 \left(\frac{d^{j-1}}{dx^{j-1}} (x^2-1)^j\right)\left(\frac{d^{j+1}}{dx^{j+1}} (x^2-1)^j\right) \mathrm{d}x \end{align*}
as you can see I am going to continue integrating by part until one of the two polynomial cancel out. The problem is how to prove rigourously all the steps. Normally I just put dots : "$\ldots$" just to say : we continue doing the same thing until we get to an easy integral.
But this is not rigourous at all. An idea to make it rigourous is to do an induction, but the problem is that induction is long and a bit cumbersome, that's why is there a way to make the above argument rigourous without using induction ?
For example, if we can express multiple integration by parts as a sum it would clearly make the above argument rigourous.
Thank you !