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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 the antiderivative of f and of f −1).
In calculus, the power rule is used to differentiate functions of the form () =, whenever is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule.
Otherwise, a function is an antiderivative of the zero function if and only if it is constant on each connected component of (those constants need not be equal). This observation implies that if a function g : U → C {\displaystyle g:U\to \mathbb {C} } has an antiderivative, then that antiderivative is unique up to addition of a function which ...
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 ...
This directly results from the fact that the integrand e −t 2 is an even function (the antiderivative of an even function which is zero at the origin is an odd function and vice versa).
In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation , and can loosely be thought of as using the chain rule "backwards."
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).
The formula for an integration by parts is () ′ = [() ()] ′ (). Beside the boundary conditions , we notice that the first integral contains two multiplied functions, one which is integrated in the final integral ( g ′ {\displaystyle g'} becomes g {\displaystyle g} ) and one which is differentiated ( f {\displaystyle f} becomes f ...