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Let I = ∫₀¹ (sin x)/√x dx and J = ∫₀¹ (cos x)/√x dx. Then which one of the following is true?

1. I > 2/3 and J > 2
2. I < 2/3 and J < 2
3. I < 2/3 and J > 2
4. I > 2/3 and J < 2

Correct answer: I < 2/3 and J < 2

Solution

Since sin x < x on (0,1), sin x/sqrt(x) < sqrt(x), so I < integral of sqrt(x) = 2/3. Since cos x < 1, cos x/sqrt(x) < 1/sqrt(x), so J < integral of 1/sqrt(x) = 2. Hence I < 2/3 and J < 2.

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