$${\int }_a^{b}f(x)\mathrm{d}x={\int }_a^{b}f(a+b-x)\mathrm{d}x$$

e.g.

• $$\begin{array}{rlrl}{\displaystyle I}& ={\int }_0^{\pi }xf(\sin x)\mathrm{d}x& & ={\int }_0^{\pi }(\pi -x)f(\sin (\pi -x))\\ & ={\int }_0^{\pi }\pi f(\sin x)-I\\ & ⟹I={\displaystyle \frac{\pi }{2}}{\int }_0^{\pi }f(\sin x)\end{array}$$