Hyperbolic logarithm

Hyperbolic logarithm

It is known that f(x) = x has an algebraic antiderivative except in the case p = –1 corresponding to the quadrature of the hyperbola. The other cases are given by Cavalieri's quadrature formula. Whereas quadrature of the parabola had been accomplished by Archimedes in the third century BC (in The Quadrature of the Parabola), the hyperbolic quadrature required the invention in 1647 of a new function: Gregoire de Saint-Vincent addressed the problem of computing the areas bounded by a hyperbola. His findings led to the natural logarithm function, once called the hyperbolic logarithm since it is obtained by integrating, or finding the area, under the hyperbola.