Let I = _0^1 x/√(x) dx and J = _0^1 x/√(x) dx . Then which one of the following is true? - Mathematics MCQ AIEEE 2026 | ExamTest.live

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1 mark

Let I = \int_{0}^{1} \frac{s i n x}{x} d x and J = \int_{0}^{1} \frac{c o s x}{x} d x . Then which one of the following is true?

Solution & Step-by-step Explanation

For 0 < x < 1, we know that sin x < x . I = \int_{0}^{1} \frac{sin x}{x} d x < \int_{0}^{1} \frac{x}{x} d x = \int_{0}^{1} x d x I < [\frac{2}{3} x^{3/2}]_{0}^{1} = \frac{2}{3} Also, for 0 < x < 1, we know cos x < 1 . J = \int_{0}^{1} \frac{cos x}{x} d x < \int_{0}^{1} \frac{1}{x} d x = [2 x]_{0}^{1} = 2 Thus, I < 2/3 and J < 2 .

Try it yourself before checking the explanation above.

Let I = \int_{0}^{1} \frac{s i n x}{x} d x and J = \int_{0}^{1} \frac{c o s x}{x} d x . Then which one of the following is true?

I > 2/3 and J > 2

I < 2/3 and J < 2

I < 2/3 and J > 2

I > 2/3 and J < 2

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