Question

${\int }_0^X\left\{{\int }_0^{u}f\right(t\left)dt\right\}$ is equal to

• A. ${\int }_0^x(x-u)f\left(u\right)du$
• B. ${\int }_0^{x}uf(x-u)du$
• C. $x{\int }_0^{x}f\left(u\right)du$
• D. $x{\int }_0^{x}uf(u-x)dx$

Answer 1

Answer: (A), (B)

The correct options are
A ${\int }_0^x(x-u)f\left(u\right)du$
B ${\int }_0^{x}uf(x-u)du$

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

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

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