Indefinite integral

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categorie: Matematica

nota: 9.85

nivel: Liceu

Since the derivative of a constant is zero, any constant may be added to an antiderivative and will still correspond to the same integral. Another way of stating this is that the antiderivative is a nonunique inverse of the derivative. For this reason, indefinite integrals are often written in the form

where is an arbitrary constant known as the constant of integration. Mathematic[...]
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Since the derivative of a constant is zero, any constant may be added to an antiderivative and will still correspond to the same integral. Another way of stating this is that the antiderivative is a nonunique inverse of the derivative. For this reason, indefinite integrals are often written in the form

where is an arbitrary constant known as the constant of integration. Mathematica returns indefinite integrals without explicit constants of integration. This means that, depending on the form used for the integrand, antiderivatives and can be obtained that differ by a constant (or, more generally, a piecewise constant function). It also means that Integrate[f+g, z] may differ from Integrate[f, z] + Integrate[g, z] by an arbitrary (piecewise) constant. Note that indefinite integrals defined algebraically deal with complex quantities. However, many elementary calculus textbooks write formulas such as

However, this work (and Mathematica) eschew the "real-only" definition, since inclusion of the absolute value means that the indefinite integral is no longer valid for a generic complex variable (the presence of the means the Cauchy-Riemann equations no longer can hold), and also violates the purely algebraic definition of indefinite integrals. Since physical problem involve definite integrals, it is much more sensible to stick with the usual complex/algebraic definitions of indefinite integration. In other words, while the Riemann integral
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