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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.
Thus, the integral of the velocity function (the derivative of position) computes how far the car has traveled (the net change in position). The first fundamental theorem says that the value of any function is the rate of change (the derivative) of its integral from a fixed starting point up to any chosen end point.
In complex analysis, a branch of mathematics, the antiderivative, or primitive, of a complex-valued function g is a function whose complex derivative is g.More precisely, given an open set in the complex plane and a function :, the antiderivative of is a function : that satisfies =.
For example, to obtain the antiderivative of that has the value 400 at x = π, then only one value of will work (in this case =). The constant of integration also implicitly or explicitly appears in the language of differential equations. Almost all differential equations will have many solutions, and each constant represents the unique ...
The points labelled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point. Sin( θ ) , Tan( θ ) , and 1 are the heights to the line starting from the x -axis, while Cos( θ ) , 1 , and Cot( θ ) are lengths along the x -axis starting from the origin.
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 ...
If ′ has a point of discontinuity then its antiderivative may not have a derivative at that point.) If the interval of integration is not compact, then it is not necessary for to be absolutely continuous in the whole interval or for ′ to be Lebesgue integrable in the interval, as a couple of examples (in which and are continuous and ...
This is only useful if the integral exists. In particular we need ′ to be non-zero across the range of integration. It follows that a function that has a continuous derivative has an inverse in a neighbourhood of every point where the derivative is non-zero. This need not be true if the derivative is not continuous.
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