Chapter 12.5, Problem 56PS

To determine

To show:the given integral ${\displaystyle \underset{0}{\overset{x}{\int }}}{\displaystyle \underset{0}{\overset{v}{\int }}f\left(u\right)du}dv={\displaystyle \underset{0}{\overset{x}{\int }}\left(x-u\right)f\left(u\right)du}$ and ${\displaystyle \underset{0}{\overset{x}{\int }}}{\displaystyle \underset{0}{\overset{v}{\int }}}{\displaystyle \underset{0}{\overset{u}{\int }}f\left(w\right)dwdudv}=\frac{1}{2}{\displaystyle \underset{0}{\overset{x}{\int }}{\left(x-w\right)}^{2}f\left(w\right)dw}$