Question

Assertion :${\int }_{1/3}^{11/2}\left\{x\right\}dx=\frac{185}{72}$ $($ where $\left\{x\right\}$ denotes the fractional part of $x$ Reason: If $f\left(x\right)$ is a periodic function having period, $T,$ then ${\int }_a^{b+nT}f\left(x\right)dx=n{\int }_0^{T}f\left(x\right)dx+{\int }_a^{b}f\left(x\right)dx$

• 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

We have,

${\int }_a^{b+nT}f\left(x\right)dx={\int }_a^{b+nT}f\left(x\right)dx+{\int }_{a+nT}^{a+nT}f\left(x\right)dx$ $={\int }_0^{nT}f\left(x\right)dx={\int }_a^{b}f\left(x\right)dx$

$=n{\int }_0^{T}f\left(x\right)dx+{\int }_a^{T}f\left(x\right)dx$

Now, $I={\int }_{1/3}^{11/2}\left\{x\right\}dx={\int }_{1/3}^{\frac{1}{2}+5}\left\{x\right\}dx$ $={\int }_{1/3}^{\frac{1}{3}+5}f\left(x\right)dx+{\int }_{\frac{1}{3}+5}^{\frac{1}{2}+5}\left\{x\right\}dx$

$=5{\int }_0^1\left\{x\right\}dx+{\int }_{1/3}^{1/2}xdx$ $(\because$ period of $x$ is$)$

$=5{\int }_0^{1}xdx+{\int }_{1/3}^{1/2}xdx$ $={\frac{{5x}^2}{2}}_0^1+{\frac{x^2}{2}}_{1/3}^{1/2}$

$=\frac{5}{2}+\frac{1}{2}(\frac{1}{4}-\frac{1}{9})=\frac{5.37}{2.36}=\frac{185}{72}$