Question

Assertion :If n is a positive integer then ${\int }_0^{n\pi }\left|\frac{\sin x}{x}\right|dx\ge \frac{2}{\pi }(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n})$ Reason: In the interval $(0,\frac{\pi }{2})$, $\frac{\sin x}{x}\ge \frac{2}{\pi }$

• 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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

${\int }_0^{n\pi }\left|\frac{\sin x}{x}\right|dx={\int }_0^{\pi }\left|\frac{\sin x}{x}\right|dx+{\int }_{\pi }^{2\pi }\left|\frac{\sin x}{x}\right|dx+...+{\int }_{(n-1)\pi }^{n\pi }\left|\frac{\sin x}{x}\right|dx$
$=I_1+I_2+I_3+...+I_n$(say)
Putting $x=t+\pi ,t+2\pi ,...,t+(n-1)\pi$ respectively in $I_2,I_3,...,I_n$
$={\int }_0^{\pi }\frac{\sin x}{x}dx+{\int }_0^{\pi }\left|\frac{\sin (t+\pi )}{t+\pi }\right|dt+{\int }_0^{\pi }\left|\frac{\sin (t+2\pi )}{t+2\pi }\right|dt+...$
$={\int }_0^{\pi }\frac{\sin x}{x}dx+{\int }_0^{\pi }\frac{\sin t}{t+\pi }dt+{\int }_0^{\pi }\frac{\sin t}{t+2\pi }dt+...$so on $(\because \frac{\sin x}{x}>0in(0,\pi ))$
$={\int }_0^{\pi }\frac{\sin x}{x}dx+{\int }_0^{\pi }\frac{\sin x}{x+\pi }dx+{\int }_0^{\pi }\frac{\sin x}{x+2\pi }dx+...$
(merely cbanging the variable)
$=\sum _{r=1}^n{\int }_0^{\pi }\frac{\sin x}{(n+(r-1)\pi )}>\sum _{r=1}^n{\int }_0^{\pi }\frac{\sin x}{(\pi +(r-1)\pi )}$
$=\sum _{r=1}^n{\int }_0^{\pi }\frac{\sin x}{\pi r}dx=\sum _{r=1}^n\frac{2}{\pi r}=\frac{2}{\pi }[1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}]$