Question about a proposition in Brezis' Sobolev Spaces

Question

This is a proposition in Brezis' book (Functional Analysis, Spaces and PDE), I wonder, where is this equality from?

https://i.stack.imgur.com/kmJHO.png

Ps: Sorry for using a image, I don't have practice on Latex, thanks in advance.

Answer 1

Split the integral $$\int_Iu(x+h)\phi(x)dx-\int_Iu(x)\phi(x)dx,$$ substitute $y=x+h$ in the first integral to get $\int_Iu(y)\phi(y-h)dy$ rename the variable $x$ to get $$\int_Iu(x)\phi(x-h)dx-\int_Iu(x)\phi(x)dx=\int_I u(x)(\phi(x-h)-\phi(x))dx.$$