Let I = _0^1 x x ;dx and ;J = _0^1 x x ;dx Then which one of the … - Vidyadip

MCQ

Let $I = \mathop \smallint \limits_0^1 \frac{{\sin x}}{{\sqrt x }}\;dx$ and $\;J = \mathop \smallint \limits_0^1 \frac{{\cos x}}{{\sqrt x }}\;dx$ Then which one of the following is true?

A
$I > \frac{2}{3}$ and $J > 2$
✓
$\;I < \frac{2}{3}$ and $J < 2$
C
$\;I < \frac{2}{3}$ and $J > 2$
D
$\;I > \frac{2}{3}$ and $J < 2$

✓

Answer

Correct option: B.

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

b
In the interval of integration $\sin x$

$I=\int_{0}^{1} \frac{\sin x}{\sqrt{x}} d x<\int_{0}^{1} \frac{x}{\sqrt{x}} d x$$=\int_{0}^{1} \sqrt{x} d x=\left[\frac{2}{3} x^{3 / 2}\right]_{0}^{1}=\frac{2}{3}$

$I<\frac{2}{3}$

Also $J=\int_{0}^{1} \frac{\cos x}{\sqrt{x}} d x<\int_{0}^{1} \frac{1}{\sqrt{x}} d x$$=[2 \sqrt{x}]_{0}^{1}=2$

$J<2$

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