Question

${\int }_a^{na}f\left(x\right)dx=(n+k){\int }_0^{a}f\left(x\right)dx,$ find $k$?

• A. 0
• B. 1
• C. -1
• D. none of these

Answer 1

Answer: (C)

-1

${\int }_0^{na}f\left(x\right)dx={\int }_0^{a}f\left(x\right)dx+{\int }_a^{2a}f\left(x\right)dx+.....+....+{\int }_{(n-1)a}^{na}f\left(x\right)dx$
For ${\int }_a^{2a}f\left(x\right)dx$ put $x=a+t$
${\int }_a^{2a}f\left(x\right)dx={\int }_0^{a}f(a+t)dt={\int }_0^{a}f(a+x)dx={\int }_0^{a}f\left(x\right)dx$
For ${\int }_{2a}^{3a}f\left(x\right)dx$ put $x=a+t$
${\int }_{2a}^{3a}f\left(x\right)dx={\int }_a^{2a}f(a+t)dt={\int }_a^{2a}f(a+x)dx={\int }_a^{2a}f\left(x\right)dx={\int }_0^{a}f\left(x\right)dx\therefore {\int }_0^{na}f\left(x\right)dx=n{\int }_0^{a}f\left(x\right)dx{\int }_{na}^{a}f\left(x\right)dx={\int }_a^{0}f\left(x\right)dx+{\int }_0^{na}f\left(x\right)dx=-{\int }_0^{a}f\left(x\right)dx+n{\int }_0^{a}f\left(x\right)dx$