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

The given curve is y = tan x and at x = π 4 ,   y = tan π 4 = 1 Also, d y d x = s e c 2 x and d y d x x = π 4 = s e c 2 π 4 = 2 2 = 2 We know that the equation of the tangent to a curve y = f x at a point x 1 ,   y 1 is y - y 1 = d y d x x = π 4 x - x 1 Hence, the equation of the tangent to y = tan x at P π 4 ,   1 is y - 1 = 2 x - π 4 ⇒ y - 1 = 2 x - π 2 ⇒ y - 1 + π 2 = 2 x To find the point where this line cuts the x -axis, put y = 0 , to get