The value of the integral I = ∫ 0 π x 1 + tan 6 ⁡ x d x ,   x ≠ π 2 is equal to

Step-by-step solution

Applying a + b - x and adding, we get, 2 I = ∫ 0 π x + π - x 1 + tan 6 ⁡ x d x ⇒ 2 I = π ∫ 0 π d x 1 + tan 6 ⁡ x ⇒ 2 I = π ∫ 0 π 2 1 1 + tan 6 ⁡ x + 1 1 + tan 6 ⁡ π - x d x ⇒ 2 I = 2 π ∫ 0 π 2 d x 1 + tan 6 ⁡ x = 2 π ∫ 0 π 2 cos 6 ⁡ x cos 6 ⁡ x + sin 6 ⁡ x d x Again, applying a + b - x and adding, we get, 2 I = π ∫ 0 π 2 sin 6 ⁡ x + cos 6 &#8289