Let the slope of the tangent to a curve y = f x at x , y be given by 2 tan x cos x - y . if the curve passes through the point π 4 , 0 , then the value of ∫ 0 π 2 y d x is equal to

Step-by-step solution

Given, d y d x = 2 tan x cos x - 2 tan x · y d y d x + 2 tan x y = 2 sin x Now solving linear differential equation by finding Integrating factor = e ∫ 2 tan x d x = 1 cos 2 x Solution is given by, y 1 cos 2 x = ∫ 2 sin x cos 2 x d x ⇒ y sec 2 x = 2 cos x + C ⇒ y = 2 cos x + C cos 2 x Passes through π 4 , 0 ⇒ 0 = 2 + C 2 ⇒ C = - 2 2 So, f x = 2 cos x - 2 2 cos 2 x : Required curve Now, ∫ 0 π 2 y d x = 2 ∫ 0 π 2 cos x d x - 2 2 ∫ 0 π 2 co