Let a curve y = y x be given by the solution of the differential equation cos 1 2 cos - 1 e - x d x = e 2 x - 1 d y . If it intersects y -axis at y = - 1 , and the intersection point of the curve with x - axis is α , 0 , then e α is equal to

Step-by-step solution

We have, cos 1 2 cos - 1 e - x d x = e 2 x - 1 d y     . . . i Put cos - 1 e - x = θ ,   θ ∈ 0 , π ⇒ cos θ = e - x ⇒ 2 cos 2 θ 2 - 1 = e - x ⇒ cos θ 2 = e - x + 1 2 = e x + 1 2 e x Hence, by i , we have cos θ 2 d x = e 2 x - 1 d y ⇒ e x + 1 2 e x d x = e 2 x - 1 d y ⇒ e x + 1 2 e x d x = e x - 1 e x + 1 d y ⇒ 1 2 ∫ d x e x e x - 1 = ∫ d y Put e x = t ⇒ d t = e x d x 1 2 ∫ d t e x e x e x - 1 = ∫