Chapter 4.12, Problem 20P

(a) Show that $y={\displaystyle {\int }_0^{x}f}(x-t)dt$ satisfies $(dy/dx)=f(x).$ (Hint: It is helpful to make the change of variable $x-t=u$ in the integral.)

(b) Show that $y={\displaystyle {\int }_0^x(x-u)}f(u)du$ satisfies $y^{\prime \prime }=f(x)$.

(c) Show that $y=\frac{1}{(n-1)!}{\displaystyle {\int }_0^x{(x-u)}^{n-1}}f(u)du$ satisfies $y^{(n)}=f(x)$.