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Let I = ∫ 0 1 sin ⁡ x x d x a n d J = ∫ 0 1 cos ⁡ x x d x . Then, which one of the following is true?

Options

  1. AI > 2 3 a n d J < 2
  2. BI > 2 3 a n d J > 2
  3. CI < 2 3 a n d J < 2
  4. DI < 2 3 a n d J > 2

Correct answer

C. I < 2 3 a n d J < 2

Step-by-step solution

Since, I = &#8747; 0 1 sin &#8289; x x d x &#60; &#8747; 0 1 x x d x as in x &#8712; 0,1 , &#160; x &#62; sin &#8289; x I &#60; &#8747; 0 1 x d x = 2 3 x 3 / 2 0 1 &#8658; I &#60; 2 3 For, x &#8712; 0,1 , &#160; cos &#8289; x x &#60; 1 x Hence, J = &#8747; 0 1 cos &#8289; x x d x &#60; &#8747; 0 1 x - 1 2 d x = 2 &#8658; J &#60; 2

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