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

Step-by-step solution

For x ∈ [ 0 , 1 ] , consider f ( x ) = x - sin x ⇒ f ' ( x ) = 1 - cos x ≥ 0 f ( x ) is an increasing function f ( x ) ≥ f ( 0 ) ⇒ f ( x ) > 0 ⇒ x > sin x Now, I = ∫ 0 1 sin x x d x ⇒ I < ∫ 0 1 x x d x ⇒ I < 2 3 x 3 2 0 1 ⇒ I < 2 3 Also, cos x ≤ 1 Now, J = ∫ 0 1 cos x x d x ⇒ J < ∫ 0 1 1 x d x ⇒ J < 2 x 0 1 ⇒ J < 2 .