Let $\mathrm{I}=\int_0^1 \frac{\sin \mathrm{x}}{\sqrt{\mathrm{x}}} \mathrm{dx}$ and $\mathrm{J}=\int_0^1 \frac{\cos \mathrm{x}}{\sqrt{\mathrm{x}}} \mathrm{dx}$. Then which one of the following is true?

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Aicharya · Accepted Answer

$ \begin{aligned} & \mathrm{I}=\int_0^1 \frac{\sin x}{\sqrt{x}} \mathrm{dx} < \int_0^1 \frac{\mathrm{x}}{\sqrt{\mathrm{x}}} \mathrm{dx}=\int_0^1 \sqrt{\mathrm{x}} \mathrm{dx}=\left.\frac{2}{3} \mathrm{x}^{3 / 2}ight|_0 ^1=\frac{2}{3} \ & \Rightarrow \mathrm{I} < \frac{2}{3} \ & \mathrm{~J}=\int_0^1 \frac{\cos x}{\sqrt{\mathrm{x}}} \mathrm{dx} < \int_0^1 \frac{1}{\sqrt{\mathrm{x}}} \mathrm{dx}=\left.2 \sqrt{\mathrm{x}}ight|_0 ^1=2 \ & 	herefore \mathrm{J} \leq 2 . \end{aligned} $

Let $\mathrm{I}=\int_0^1 \frac{\sin \mathrm{x}}{\sqrt{\mathrm{x}}} \mathrm{dx}$ and $\mathrm{J}=\int_0^1…

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