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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 ...
The inverse chain rule method (a special case of integration by substitution) Integration by parts (to integrate products of functions) Inverse function integration (a formula that expresses the antiderivative of the inverse f −1 of an invertible and continuous function f, in terms of f −1 and the antiderivative of f).
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
His second proof was geometric. If () = and () =, the theorem can be written: + =.The figure on the right is a proof without words of this formula. Laisant does not discuss the hypotheses necessary to make this proof rigorous, but this can be proved if is just assumed to be strictly monotone (but not necessarily continuous, let alone differentiable).
It may be used to prove Nicomachus's theorem that the sum of the first cubes equals the square of the sum of the first positive integers. [2] Summation by parts is frequently used to prove Abel's theorem and Dirichlet's test.
It can also be applied to a power series with radius of convergence R ≠ 1 by a simple change of variables ζ = z/R. [2] Notice that Abel's test is a generalization of the Leibniz Criterion by taking z = −1.
The validity of this rule follows from the validity of the Feynman method, for one may always substitute a subscripted del and then immediately drop the subscript under the condition of the rule. For example, from the identity A ⋅( B × C ) = ( A × B )⋅ C we may derive A ⋅(∇× C ) = ( A ×∇)⋅ C but not ∇⋅( B × C ) = (∇× B ...
This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where a ( x ) = a ∈ R {\displaystyle a(x)=a\in \mathbb {R} } is constant, b ( x ) = x , {\displaystyle b(x)=x,} and f ( x , t ...