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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.
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and ()
Its antiderivatives + do not seem to satisfy the requirements of the theorem, since they are not (apparently) sums of rational functions and logarithms of rational functions. However, a calculation with Euler's formula e i θ = cos θ + i sin θ {\displaystyle e^{i\theta }=\cos \theta +i\sin \theta } shows that in fact the ...
When an antiderivative of exists, then there are infinitely many antiderivatives for , obtained by adding an arbitrary constant to . Also, by the first part of the theorem, antiderivatives of f {\displaystyle f} always exist when f {\displaystyle f} is continuous.
Many of the following antiderivatives have a term of the form ln |ax + b|. Because this is undefined when x = −b / a, the most general form of the antiderivative replaces the constant of integration with a locally constant function. [1] However, it is conventional to omit this from the notation.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every interval on which f is not zero, but may be discontinuous at the points where f(x) = 0.
When taking the antiderivative, Lagrange followed Leibniz's notation: [7] = ′ = ′. However, because integration is the inverse operation of differentiation, Lagrange's notation for higher order derivatives extends to integrals as well. Repeated integrals of f may be written as
Finding an elementary antiderivative is very sensitive to details. For instance, the following algebraic function (posted to sci.math.symbolic by Henri Cohen in 1993 [3]) has an elementary antiderivative, as Wolfram Mathematica since version 13 shows (however, Mathematica does not use the Risch algorithm to compute this integral): [4] [5]