If the integral I = ∫ 0 π sec - 1 ⁡ sec ⁡ x 1 + tan 8 ⁡ x d x ,   ∀ x ≠ π 2 , then the value of I is equal to (where · is the greatest integer function)

Step-by-step solution

Applying a + b - x and adding, we get, ⇒ 2 I = π ∫ 0 π d x 1 + tan 8 ⁡ x ⇒ 2 I = π ∫ 0 π / 2 1 1 + tan 8 ⁡ x + 1 1 + tan 8 ⁡ π - x d x ⇒ 2 I = 2 π ∫ 0 π 2 d x 1 + tan 8 ⁡ x = 2 π ∫ 0 π 2 cos 8 ⁡ x cos 8 ⁡ x + sin 8 ⁡ x ⇒ I = π ∫ 0 π 2 cos 8 ⁡ x cos 8 ⁡ x + sin 8 ⁡ x Applying a + b - x and adding, 2 I = π ∫ 0 π 2 sin 8 ⁡ x + cos