Question

If $I={\int }_0^1\frac{sinx}{\sqrt{x}}dxandJ={\int }_0^1\frac{cosx}{\sqrt{x}}dx,$ then, which one of the following is true?

• A. $I>\frac{2}{3}andJ>2$
• B. $I<\frac{2}{3}andJ<2$
• C. $I<\frac{2}{3}andJ>2$
• D. $I>\frac{2}{3}andJ<2$

Answer 1

The correct option is B $I<\frac{2}{3}andJ<2$

$Since,I={\int }_0^1\frac{sinx}{\sqrt{x}}dx<{\int }_0^1\frac{x}{\sqrt{x}}dx,$
$(\because xϵ(0,1),x>sinx)$
$\Rightarrow I<{\int }_0^1\sqrt{x}dx=\frac{2}{3}{\left[X^{\frac{3}{2}}\right]}_0^1$
$\Rightarrow I<\frac{2}{3}$
$andJ={\int }_0^1\frac{cosx}{\sqrt{x}}dx<{\int }_0^1{x}^{-\frac{1}{2}}dx=2$
$\Rightarrow J<2$