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?

(B) I $<\frac{2}{3}$ and J $<2$

(A) I $>\frac{2}{3}$ and J $>2$

(D) I $>\frac{2}{3}$ and J $<2$

JEE Main · 2008 · 27 Apr 2008 · Offline · Definite Integrals

Let

I = \int_{0}^{1} \frac{s i n x}{x} dx

and

J = \int_{0}^{1} \frac{c o s x}{x} dx

. Then which one of the following is true?

Answer: (B)

On the interval

(0, 1]

one has

sin x < x

and

cos x < 1

. Hence

I = \int_{0}^{1} \frac{s i n x}{x} d x < \int_{0}^{1} \frac{x}{x} d x = \int_{0}^{1} x d x = \frac{2}{3}

, and

J = \int_{0}^{1} \frac{c o s x}{x} d x < \int_{0}^{1} \frac{1}{x} d x = 2

. So

I < \frac{2}{3}

and

J < 2

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option