Area hyperbolic tangent
If we consider the integral
it would be nice to know where it came from. It can’t be a derivative of because we miss in the numerator. Could it originate from another known function? Let’s add in the denominator and let’s immediately subtract so nothing changes. We get
Now we substitute
It reminds us of
Let’s make four times larger to get rid of the .
and we’re in the home stretch:
We revert to with and so
Manhandling a circle will leave you with an ellipse. Manhandling an ellipse will leave you with a parabola. Manhandling a parabola will leave you with a hyperbola. They are all all the result of an intersection with a cone.
So let’s look at
and then at
That gives us two points with asymptotes from here to infinity: and . But tilt the hyperbola far enough in 3D and you will get close to your your original . Would that mean tilting in the cone as well? That would mean we would go from a hyperbolic function to some trigonometric function of a circle or ellipse, like , or .
That reminds us of
follow the same steps and without further ado we would end up with
From to . From hyperbola via parabola to ellipse and circle.