TITLE: Why does 'The King Property' of integration work?
QUESTION [9 upvotes]: Today I learned about something known as the king's property which really helps in solving integrals and I wanted to know why does this property work.
I dont know if this terminology is used elsewhere ,
so what im talking about is this property
$$ \int_a^b f(x)dx = \int_a^b f(a+b-x)dx $$

REPLY [6 votes]: If $x=a+b-y$ then $$\int_{x=a}^{x=b}f(x)dx=\int_{y=b}^{y=a}f(a+b-y)d(a+b-y)=$$ $$=\int_{y=b}^{y=a}f(a+b-y)(-dy)=$$ $$=-\int_{y=b}^{y=a}f(a+b-y)dy=$$ $$=\int_{y=a}^{y=b}f(a+b-y)dy.$$  In the last integral we can replace each "$y$" with an "$x$" to get $\int_{x=a}^{x=b}f(a+b-x)dx.$ This is a different and separate change-of-variable, and is $not$ the earlier "$x=a+b-y$".