Question

Let $I={\int }_0^{\frac{\pi }{6}}\frac{cosx}{x}dx,J={\int }_{\frac{\pi }{3}}^{\frac{\pi }{2}}\frac{cosx}{x}dx.$ Which of the following is correct?

• A. $I<\frac{\pi }{6},J<\frac{\pi }{6}$
• B. $I>\frac{\pi }{6},J<\frac{\pi }{6}$
• C. $I<\frac{\pi }{6},J>\frac{\pi }{6}$
• D. $I>\frac{\pi }{6},J>\frac{\pi }{6}$

Answer 1

The correct option is B

$I>\frac{\pi }{6},J<\frac{\pi }{6}$

$0<x<\frac{\pi }{6}cosx>x\Rightarrow \frac{cosx}{x}>1{\int }_0^{\frac{\pi }{6}}\frac{cosx}{x}dx>{\int }_0^{\frac{\pi }{6}}1.dx\text{(Area under the curve)}\therefore I>\frac{\pi }{6}\frac{\pi }{3}<x<\frac{\pi }{2}cosx<x\Rightarrow \frac{cosx}{x}<1{\int }_{\frac{\pi }{3}}^{\frac{\pi }{2}}\frac{cosx}{x}dx<{\int }_{\frac{\pi }{3}}^{\frac{\pi }{2}}1.dx\text{(Area under the curve)}\therefore J<\frac{\pi }{6}$