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Given a real function f, any function φ such that φ′(x)=f(x), for all x (in the domain of f), is an antiderivative of f. If φ1 and φ2 are both antiderivatives of a continuous function f, then φ1(x) and φ2(x) differ by a constant. In that case, the notation

f(x) dx

may be used for an antiderivative of f, with the understanding that an arbitrary constant can be added to any antiderivative. Thus,

f(x) dx + c,

where c is an arbitrary constant, is an expression that gives all the antiderivatives.

Subjects: Mathematics.

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