In this problem, we are given the eccentricity of a hyperbola and we need to find the integration of its conjugate hyperbola. Let's begin by understanding the terms in detail.


A hyperbola is a conic section formed by the intersection of a plane with a double cone. It has two parts - the left branch and the right branch. The distance between the two branches is called the transverse axis, and the distance between the two vertices on the transverse axis is called the length of the transverse axis. The eccentricity of a hyperbola is defined as the ratio of the distance between the two foci to the length of the transverse axis.


The conjugate hyperbola of a hyperbola is the hyperbola whose transverse axis is perpendicular to the transverse axis of the given hyperbola. It is obtained by interchanging the roles of x and y in the equation of the given hyperbola.


Now, let's move on to the integration part. We are given the integration of f(e) f(f(e)) from 1 to √2, where f(e) is the conjugate hyperbola of the given hyperbola with eccentricity e.


We can simplify this integration by using the substitution t = f(e). This gives us the limits of integration as f(1) and f(√2).


Using the equation of the conjugate hyperbola, we can express f(e) in terms of e as f(e) = √(e^2 - 1). Similarly, we can express f(f(e)) in terms of e as f(f(e)) = √(1 - e^2).


Substituting these values in the integration, we get ∫(√(e^2 - 1) * √(1 - e^2)) de from 1 to √2. We can simplify this using the substitution e = sinθ, which gives us de = cosθ dθ and the limits of integration as 0 and π/4.


Substituting these values in the integration, we get ∫(cosθ * sinθ) dθ from 0 to π/4, which simplifies to 1/4. Therefore, the final result of the integration is 1/4.


In this problem, we used the concept of hyperbola and conjugate hyperbola to evaluate the integration of f(e) f(f(e)) from 1 to √2. We simplified the integration using the substitution t = f(e) and then used the equation of the conjugate hyperbola to express f(e) and f(f(e)) in terms of e. Finally, we evaluated the integration using the substitution e = sinθ and obtained the final result as 1/4.