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A function F(x) is an h-antiderivative of f(x) if D h F(x) = f(x).The h-integral is denoted by ().If a and b differ by an integer multiple of h then the definite integral () is given by a Riemann sum of f(x) on the interval [a, b], partitioned into sub-intervals of equal width h.
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 continuous function f is a differentiable function F whose derivative is equal to the original function f.
However, this formal similarity notwithstanding, possessing a complex-antiderivative is a much more restrictive condition than its real counterpart. While it is possible for a discontinuous real function to have an anti-derivative, anti-derivatives can fail to exist even for holomorphic functions of a complex variable.
The x antiderivative of y and the second antiderivative of f, Euler notation. D-notation can be used for antiderivatives in the same way that Lagrange's notation is [ 11 ] as follows [ 10 ] D − 1 f ( x ) {\displaystyle D^{-1}f(x)} for a first antiderivative,
Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed ...
Q-differentiation resembles ordinary differentiation, with curious differences.For example, the q-derivative of the monomial is: [2] = = []where [] is the q-bracket of n.Note that [] = so the ordinary derivative is regained in this limit.
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The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals.