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Let J = ∫ from 0 to 1 of (cos x)/√x dx and K = ∫ from 0 to 1 of (sin x)/√x dx. Which of the following statements is correct?

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

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

Solution

K=integral_0^1 (sin x)/sqrt(x) dx ~ 0.62 < 2/3, and J=integral_0^1 (cos x)/sqrt(x) dx ~ 1.81 < 2. Hence K < 2/3 and J < 2.

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