Question

Assertion :${\int }_0^{100}(x-\left|x\right|)dx=50$, where [x] denotes greatest integer function. Reason: If $f\left(x\right)$ is a periodic function with period $\tau$, then ${\int }_0^{n\tau }f\left(x\right)dx=n{\int }_0^{\tau }f\left(x\right)dx$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

${\int }_0^{\pi \tau }f\left(x\right)dx=\sum _{r=1}^n{\int }_{(r-1)\tau }^{r\tau }f\left(x\right)dx$

$={\sum }_{r=1}^n{\int }_0^{\tau }f((r-1)\tau +y)dy$, $x=(r-1)\tau +y$

$={\sum }_{r=1}^n{\int }_0^{\tau }f\left(y\right)dy$

(as f(x) is periodic with period $\tau$ $\therefore f((r-1)\tau +y)=f\left(y\right)$)

$=n{\int }_0^{\tau }f\left(y\right)dy=n{\int }_0^{\tau }f\left(x\right)dx$ (merely changing the variable)

$\therefore$ Reason (R) is true

Now ${\int }_0^{100}(x-\left[x\right])dx={\int }_0^{100\left(1\right)}(x-\left[x\right])dx$

$=100{\int }_0^1(x-\left[x\right])dx$ ($\because$ period of x-[x] is 1)

$=100{\int }_0^{1}xdx=50$

$\therefore$ Assertion (A) is followed by Reason (R).