Question

Assertion :Statement 1 If $n$ is 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: Statement 2 $\frac{\sin x}{x}\ge \frac{2}{\pi }$ on $(0,\pi /2)$

• 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

${\int }_0^{n\pi }\left|\frac{\sin x}{x}\right|dx={\int }_0^{\pi }\frac{\sin x}{x}dx+{\int }_{\pi }^{2\pi }\frac{\left|\sin x\right|}{x}dx+....+{\int }_{(n-1)\pi }^{n\pi }\frac{\left|\sin x\right|}{x}dx$
${\int }_0^{\pi }\frac{\sin x}{x}dx+{\int }_0^{\pi }\frac{\sin u}{\pi +u}du+{\int }_0^{\pi }\frac{\sin u}{\pi +2u}du+{\int }_0^{\pi }\frac{\sin u}{u+(n-1)\pi }du$
(Putting $x-\pi =u$ in 2nd integral, $x-2\pi =u$ in $3rd$ integtral...,)
${\int }_0^{\pi }\frac{\sin u}{u+(r-1)\pi }\ge {\int }_0^{\pi }\frac{\sin u}{\pi +(r-1)\pi }=\frac{2}{r\pi }r=1,2,...,n$
$\therefore {\int }_0^{n\pi }\left|\frac{\sin x}{x}\right|dx\ge \frac{2}{\pi }[1+\frac{1}{2}+...+\frac{1}{n}]$
Statement 2 is true but is not a correct reason for the statement- 1