Question

If $f(a+b-x)=f\left(x\right),$ then ${\int }_a^{b}xf\left(x\right)dx$ is equal to :

Answer 1

$\begin{array}{c}wehave\\ I={\int }_a^{b}xf\left(x\right)dx-----\left(1\right)\\ Putx=a+b-x\\ d\left(a+b-x\right)=-dx\\ NOw,\\ I={\int }_a^b-\left(a+b-x\right)f\left(a+b-x\right)dx-----\left(2\right)\\ ={\int }_a^b\left(a+b-c\right)f\left(x\right)dx\\ nowbyadding\left(1\right)+\left(2\right)\\ 2I=\left(a+b\right){\int }_a^{b}f\left(x\right)dx\\ I=\frac{\left(a+b\right)}{2}{\int }_a^{b}f\left(x\right)dx\\ whichistherequiredanswer.\end{array}$