Using what basis? That's a continuous function and all continuous functions are integrable.
If you do not know that theorem, then the simplest way to show a function is "integrable" is to find its integral!
If x< 1/2, |1- 2x|= 1- 2x. Integrate that, including the "constant of integration" C. If x> 1/2, |1- 2x|= -(1- 2x)= 2x- 1. Integrate that, including the "constant of integration" D. At x= 1/2, those must be the same, allowing you to write either C in terms of D or D in terms of C so that you have only a single "constant of integration".