Dear friends,

I have a problem on multiple integrals.

Now, I explain it below:

Let $\displaystyle x=[x_{1},\ldots,x_{n}]$ be a vector, $\displaystyle E:=[a_{1},b_{1}]\times\ldots\times[a_{n},b_{n}]$ be a rectangle, where $\displaystyle a_{i},b_{i}$ are reals for $\displaystyle i=1,\ldots,n$, and $\displaystyle u:E\to\mathbb{R}$ be a continuous function, which vanishes on the boundary $\displaystyle \partial E.$

Then, the following is written

$\displaystyle \int\limits_{E}\bigg\{\big[(x_{i}-a_{i})(b_{i}-x_{i})\big]\int\limits_{a_{i}}^{b_{i}}\bigg|\frac{\partial}{\ partial s_{i}}u(x;s_{i})\bigg|d s_{i}\bigg\}dx$

$\displaystyle =$

$\displaystyle

\Bigg(\int\limits_{a_{i}}^{b_{i}}\big[(x_{i}-a_{i})(b_{i}-x_{i})\big]d x_{i}\Bigg)\Bigg(\int\limits_{E}\bigg|\frac{\parti al}{\partial x_{i}}u(x)\bigg|dx\Bigg)$

for $\displaystyle i=1,\ldots,n$, where $\displaystyle u(x;s_{i}):=u(x_{1},\ldots,x_{i-1},s_{i},x_{i+1},\ldots,x_{n}).$

How can we get the second line from the first one?

By partial integration (but how)?