\(\int_a^b f(x) d x\) is equal to :

Let \(x=a+b-u\).

\(dx=-du.\)

When \(x=a, u=b\);

when \(x=b, u=a\).

\({\int }_{a}^{b}f\left(x\right)dx={\int }_{u=b}^{u=a}f\left(a+b-u\right)\left(-du\right)\)

\(\Rightarrow {\int }_{a}^{b}f\left(x\right)dx={\int }_{u=a}^{u=b}f\left(a+b-u\right)du\)

Rename \(u\) to \(x\) :

\(\int_a^b f(x) d x=\int_a^b f(a+b-x) d x\).