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Calculus. 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."
Nonelementary integral. In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field ...
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a function f is a differentiable function F whose derivative is equal to the original function f.
Morera's theorem states that a continuous, complex -valued function f defined on an open set D in the complex plane that satisfies for every closed piecewise C1 curve in D must be holomorphic on D . The assumption of Morera's theorem is equivalent to f locally having an antiderivative on D . The converse of the theorem is not true in general.
Constant of integration. In calculus, the constant of integration, often denoted by (or ), is a constant term added to an antiderivative of a function to indicate that the indefinite integral of (i.e., the set of all antiderivatives of ), on a connected domain, is only defined up to an additive constant. [1] [2] [3] This constant expresses an ...
In mathematics, Liouville's theorem, originally formulated by French mathematician Joseph Liouville in 1833 to 1841, [1] [2] [3] places an important restriction on antiderivatives that can be expressed as elementary functions . The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions.
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Miscellanea. v. t. e. In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rational function of by setting . This is the one-dimensional stereographic projection of the unit circle parametrized by angle ...