An indefinite integral with no constant of integration

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Discussion Overview

The discussion revolves around the indefinite integral of the form f'(x)/f(x) and the implications of the constant of integration in such integrals. Participants explore various examples and counterexamples, questioning the necessity and value of the constant of integration in different contexts, including real and complex functions.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the indefinite integral of f'(x)/f(x) is log(|f(x)|) without the need for a constant of integration, suggesting that the original constant in the function is recovered.
  • Others argue that while a constant added to a function does not affect its behavior in differential equations, it still alters the function's value and can be significant.
  • A participant presents an example where the integral of log(-A*x) leads to a conclusion that a multiplicative constant of integration is necessary, indicating that the original constant A is not recovered.
  • Another participant clarifies that the integral of f'(x)/f(x) is indeed log(|f(x)|) + C, emphasizing that C is not necessarily zero and that the ambiguity lies in the sign of f(x) rather than the constant.
  • Some participants challenge the correctness of claims regarding the derivatives of log(|f(x)|) and the implications for the integral, leading to further debate on the validity of examples presented.
  • Confusion arises regarding the notation and meaning of g and g(x), with participants discussing the implications of treating them interchangeably in the context of integration.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the necessity of the constant of integration in indefinite integrals, with no consensus reached on whether it can be universally considered zero or if it always needs to be included. The discussion remains unresolved with differing interpretations and examples presented.

Contextual Notes

Participants note limitations in their examples, such as potential confusion between function notation and values, and the dependence on specific forms of functions being integrated. Some mathematical steps remain unresolved, particularly regarding the treatment of constants in various integrals.

  • #31
What you have is a differential equation. To find a unique solution of a first-order differential equation, you need an initial condition, say m(a) = ma.

Before continuing I should add that m(x) really represents the mass density of your rod. The actual mass of a very short section of length Δx would be m(x)Δx.

Your problem, with the addition of an initial condition, is
$$\frac{m'}{m} = \frac{1}{x}, m(a) = m_a$$

From this we get ln(m) = ln(x) + C. Exponentiating both sides, we get m = eln(x) + C = eC eln(x) = Kx, where K = eC.

So m(x) = Kx.

From the initial condition, m(a) = ma = Ka ##\Rightarrow## K = ma/a.

The unambiguous solution is m(x) = ma/a * x. You can verify that this function satisfies the differential equation and initial condition.

To find the mass of the rod in the interval [a, b], calculate integral of m(x) between a and b.
 

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