First slide

Methods of integration

Question

Let $I = \int_{0}^{1} \frac{\sin x}{\sqrt{x}} d x$ and $J = \int_{0}^{1} \frac{\cos x}{\sqrt{x}} d x$ Then which one of the following is true

Moderate

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 $J < 2$

Solution

Since $\sin x < x$ ,for $x > 0$ , we get
$\frac{\sin x}{\sqrt{x}} < \sqrt{x}$ for $0 < x < 1$
$∴ I = \int_{0}^{1} \frac{\sin x}{\sqrt{x}} d x < \int_{0}^{1} \sqrt{x} d x = \frac{2}{3}$
$J = \int_{0}^{1} \frac{\cos x}{\sqrt{x}} d x < \int_{0}^{1} \frac{1}{\sqrt{x}} d x = 2$

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