If the integral I = ∫ - sin ⁡ x x - ln ⁡ x cos ⁡ x d x = f x + C (where, C is the constant of integration) and f e = - sin ⁡ e , then the number of natural numbers less than f π 6 is equal to (where, · is the greatest integer function)

Question

If the integral I = ∫ - sin ⁡ x x - ln ⁡ x cos ⁡ x d x = f x + C (where, C is the constant of integration) and f e = - sin ⁡ e , then the number of natural numbers less than f π 6 is equal to (where, · is the greatest integer function),

MARKS App · Accepted Answer

Given Integral is I = - ∫ d sin ⁡ x ⋅ ln ⁡ x = - sin ⁡ x ⋅ ln ⁡ x + C ∴ f x = - sin ⁡ x ⋅ ln ⁡ x ⇒ f π 6 = - 1 2 ln ⁡ π 6 = 1 2 ln ⁡ 6 π ⇒ f π 6 ∈ 0 , 1 ⇒ f π 6 = 0

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