Question

state whether given statement is true or false??

${\int }_0^x\left({\int }_0^{u}f\right(t\left)dt\right)$ $du={\int }_0^{x}f\left(u\right).(x-u)du$

• A. True
• B. False

Answer 1

The correct option is A True

${\int }_o^x\left({\int }_0^{4}f\right(t\left)dt\right)du={\int }_0^{x}1.\left({\int }_0^{u}f\right(t\left)dt\right).du$

$=\left[u{\int }_0^{u}f\right(t)dt{]}_0^x-{\int }_0^x[u.\left(f\right(u).\frac{d}{du}u-f(0).\frac{d}{du}(0\left)\right)]du$ [Integrating by parts]

$=x{\int }_0^{x}f\left(t\right)dt-{\int }_0^{x}u.f\left(u\right).du=x.{\int }_0^{x}f\left(u\right)du-{\int }_0^{x}u.f\left(u\right)du=x{\int }_0^{x}f\left(u\right)du-{\int }_0^{x}uf\left(u\right)du[\because {\int }_0^{b}f(x)dx={\int }_a^{b}f(z)dz$

$={\int }_0^{x}f\left(u\right)(x-u)du$

$\therefore$ The statement is true.