If the area covered by y = 2 x and y = 2 2 x - 1 from x = 1 to x = e is ln ⁡ a sq. units, then 2 e - 1 2 a 2 is equal to

Step-by-step solution

As 2 x > 2 2 x - 1   ∀ x ∈ 1 , e ∴ the required area is A = ∫ 1 e 2 x - 2 2 x - 1 d x = 2 ln ⁡ x 1 e - 1 2 ln ⁡ 2 x - 1 1 e = ln ⁡ e 2 - ln ⁡ 2 e - 1 = ln ⁡ e 2 2 e - 1 sq. units ⇒ ln ⁡ e 2 2 e - 1 = ln a ⇒ a 2 e - 1 = e 2 ⇒ a 2 2 e - 1 2 = e 4