Let $I = \mathop \smallint \nolimits_0^1 \,\,\frac{{\sin \,\,x}}{{\sqrt x }}\,\,dx\,\,{\rm{and}}\,\,J = \mathop \smallint \nolimits_0^1 \,\,\frac{{\cos \,\,x}}{{\sqrt x }}\,\,dx$. Then, which one of the following is true?

$I < \frac{2}{3}\,\,{\rm{and}}\,\,J < 2$

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

$I > \frac{2}{3}\,\,{\rm{and}}\,\,J > 2$

$I\,\,\left\langle {\,\frac{2}{3}\,\,{\rm{and}}\,\,J\,} \right\rangle \,\,2$

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

1 Easy

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Analyzing +4 | -1

Let

I = \int \nolimits_0^1 \,\,\frac{{\sin \,\,x}}{{\sqrt x }}\,\,dx\,\,\mathrm{and}\,\,J = \int \nolimits_0^1 \,\,\frac{{\cos \,\,x}}{{\sqrt x }}\,\,dx

. Then, which one of the following is true?

View answer and explanation

Correct option: C

I < \frac{2}{3}\,\,\mathrm{and}\,\,J < 2

Option C is correct because, for x in (0,1), sin(x) < x and cos(x) < 1, allowing us to bound the integrals I and J, resulting in I < 2/3 and J < 2.