Integral equivalence/log function shifted

hnh

Hello, I am trying to show that given a 1-lipschitz (and absolutely continuous) function g which vanishes outside compact interval [-a,b] that

\int_[-a,o] g'(x) ((b-x)ln(b-x)-(b-x))dx + \int_[0,b] g'(x)((x+a)ln(x+a)-(x+a))dx

= - \int_[-a,b] g'(x)(xlnx-x)dx

Also, g' is negative for positive x and positive for negative x.

Any suggestions are helpful. Maybe someone could suggest criteria for this to hold-ie, for even functions. Or maybe graphing it would show this-I do not have maple. Maybe it is true just for ln?...

Thanks