If I 1 = ∫ 0 π 2 e sin ⁡ x 1 + x cos ⁡ x d x and I 2 = ∫ 0 π 2 e cos ⁡ x 1 - x sin ⁡ x d x , then I 1 I 2 is equal to (where x denotes the greatest integer less than or equal to x )

Step-by-step solution

I 1 = ∫ 0 π 2 e sin ⁡ x + x e sin ⁡ x . cos ⁡ x d x = x e sin ⁡ x π 2 0 = π e 2 I 2 = ∫ 0 π 2 e cos ⁡ x + x - e cos ⁡ x . sin ⁡ x d x = x e cos ⁡ x π 2 0 = π 2 Hence, I 1 I 2 = e ⇒ e = 2