The area of the region (in square units) above the x -axis bounded by the curve y ⁡ = tan x , 0 ≤ x ⁡ ≤ π 2 and the tangent to the curve at x ⁡ = π 4 is

Step-by-step solution

Required Area = ∫ 0 π 4 tan x d x - A r e a   o f   t r i a n g l e   f o r m e d   b y   t a n g e n t ,   x = π 4   a n d   x - a x i s = log ⁡ sec π 4 - log ⁡ sec 0 - 1 2 × 1 π 4 - x 0 = log ⁡ 2 - 1 4 (where x 0 = π 4 - 1 2 is x -interept of tangent) = 1 2 log ⁡ 2 - 1 4