Question

Let $l={\int }_0^1\frac{\sin x}{\sqrt{x}}dx$ and $j={\int }_0^1\frac{\cos x}{\sqrt{x}}dx$. Then which one of the following is true?

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

Answer 1

The correct option is D $I<\frac{2}{3}$ and $J<2$

We know that $\frac{\sin x}{x}<1$, when $x\in (0,1)=>\frac{\sin x}{\sqrt{x}}<\sqrt{x}$
$\therefore I={\int }_0^1\frac{\sin x}{\sqrt{x}}<{\int }_0^1\sqrt{x}dx=\frac{2}{3}$
Again
$\frac{\cos x}{\sqrt{x}}<\frac{1}{\sqrt{x}}$ when
$x\in (0,1)\therefore J={\int }_0^1\frac{cosx}{\sqrt{x}}<{\int }_0^1\frac{1}{\sqrt{x}}dx=2$