Mathematics - Definite Integration Question with Solution | TestHub

$I={\int }_a^{b}xf(x)dx={\int }_a^b(a+b-x)f(a+b-x)dx$ $=(a+b){\int }_a^{b}f(a+b-x)dx-{\int }_a^{b}xf(a+b-x)dx$ $=(a+b){\int }_a^{b}f(a+b-x)dx-{\int }_a^{b}xf(x)dx$ $2I=(a+b){\int }_a^{b}f(x)dx$ $I=\frac{(a+b)}{2}{\int }_a^{b}f(x)dx;I=\frac{(a+b)}{2}{\int }_a^{b}f(a+b-x)dx$