# Let $\dpi{100} I=\int_{0}^{1}\frac{\sin x}{\sqrt{x}}dx\; and\; J=\int_{0}^{1}\frac{\cos x}{\sqrt{x}}dx$. Then which one of the following is true? Option 1) $I> \frac{2}{3}\; and\; J< 2$ Option 2) $I> \frac{2}{3}\; and\; J> 2$ Option 3) $I< \frac{2}{3}\; and\; J< 2$ Option 4) $I< \frac{2}{3}\; and\; J> 2$

As we learnt in

Properties of Definite Integration -

If $f(x)\geqslant g(x)$ for

all $x\in \left [ a,b \right ]$ then

$\int_{a}^{b}f\left ( x \right )dx\geqslant \int_{a}^{b}g(x)dx$

-

$I_{1}=\int_{0}^{1}\frac{sinx}{\sqrt{x}}dx<\int_{0}^{1}\frac{x}{\sqrt{x}}dx=\int_{0}^{1}\sqrt{x}dx$

So, $I_{1}<\frac{2}{3}$

Now, $J = \int_{0}^{1}\frac{cosx}{\sqrt{x}}dx\leq \int_{0}^{1}\frac{1}{\sqrt{x}}dx=2$

$J\leq 2$

Option 1)

$I> \frac{2}{3}\; and\; J< 2$

Incorrect

Option 2)

$I> \frac{2}{3}\; and\; J> 2$

Incorrect

Option 3)

$I< \frac{2}{3}\; and\; J< 2$

Correct

Option 4)

$I< \frac{2}{3}\; and\; J> 2$

Incorrect

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