Editing Power rule (section)

== History==
The power rule for integrals was first demonstrated in a geometric form by Italian mathematician [[Bonaventura Cavalieri]] in the early 17th century for all positive integer values of <math>{\displaystyle n}</math>, and during the mid 17th century for all rational powers by the mathematicians [[Pierre de Fermat]], [[Evangelista Torricelli]], [[Gilles de Roberval]], [[John Wallis]], and [[Blaise Pascal]], each working independently. At the time, they were treatises on determining the area between the graph of a rational power function and the horizontal axis. With hindsight, however, it is considered the first general theorem of calculus to be discovered.<ref name="Boyer">{{cite book|last1=Boyer|first1=Carl|title=The History of the Calculus and its Conceptual Development|date=1959|publisher=Dover|location=New York|isbn=0-486-60509-4|page=[https://archive.org/details/historyofcalculu00boye/page/127 127]|url=https://archive.org/details/historyofcalculu00boye/page/127}}</ref> The power rule for differentiation was derived by [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]], each independently, for rational power functions in the mid 17th century, who both then used it to derive the power rule for integrals as the inverse operation. This mirrors the conventional way the related theorems are presented in modern basic calculus textbooks, where differentiation rules usually precede integration rules.<ref>{{cite book|last1=Boyer|first1=Carl|title=The History of the Calculus and its Conceptual Development|date=1959|publisher=Dover|location=New York|isbn=0-486-60509-4|pages=[https://archive.org/details/historyofcalculu00boye/page/191 191, 205]|url=https://archive.org/details/historyofcalculu00boye/page/191}}</ref>

Although both men stated that their rules, demonstrated only for rational quantities, worked for all real powers, neither sought a proof of such, as at the time the applications of the theory were not concerned with such exotic power functions, and questions of convergence of infinite series were still ambiguous.

The unique case of <math>r = -1</math> was resolved by Flemish Jesuit and mathematician [[Grégoire de Saint-Vincent]] and his student [[Alphonse Antonio de Sarasa]] in the mid 17th century, who demonstrated that the associated definite integral,
:<math>\int_1^x \frac{1}{t}\, dt</math>
representing the area between the rectangular hyperbola <math>xy = 1</math> and the x-axis, was a logarithmic function, whose base was eventually discovered to be the transcendental number [[e (mathematical constant)|e]]. The modern notation for the value of this definite integral is <math>\ln(x)</math>, the natural logarithm.