Understanding Integrals: Multiplying by a Constant Within the Integral?

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Discussion Overview

The discussion revolves around the manipulation of integrals, specifically the implications of multiplying a constant or variable within an integral. Participants explore whether such multiplication affects the integral's result and clarify the nature of the terms involved.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions if multiplying the left-hand side of the integral by a constant results in the right-hand side being multiplied by the same constant within its integral.
  • Another participant seeks clarification on whether the discussion pertains to multiplying a constant with the integral itself.
  • A different participant emphasizes that the multiplication involves another variable, not a constant.
  • One participant asserts that a specific relationship involving integrals and functions does not hold true, providing examples to illustrate their point.
  • Another participant references a calculus textbook section related to first-order differential equations, implying a connection to the topic at hand.

Areas of Agreement / Disagreement

Participants express differing interpretations of the original question, leading to multiple competing views on the manipulation of integrals. The discussion remains unresolved regarding the implications of multiplying terms within integrals.

Contextual Notes

There is uncertainty regarding the definitions of terms used in the discussion, particularly whether the multiplication involves constants or variables, which may affect the interpretations of the integral manipulations.

Fritz
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if you have [tex]\int_frac{y}{x}=int_L[/tex], if you multiply the LHS by c within the integral, is L multiplied by c within its integral?
 
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Do you mean this?

[tex]\int \frac{y}{x} = \int L[/tex]

- Warren
 
or are you talking about multiplying some constant with an integral ?

- Marlon

[tex]c * \int\frac {y}{x} = \int cL[/tex] ?
 
Not a constant, another variable.
 
If you mean "Is [itex]f(x)\int g(x)dx= \int f(x)g(x)dx[/itex]" the answer is no:
[itex]x\int xdx= \frac{1}{2}x^3+ C[/itex] and [itex]\int x^2dx= \frac{1}{3}x^3+ C[/itex].

I may be misunderstanding your question since if you are taking differential equations, you certainly should know calculus enough that that would be obvious.
 
Stewart has a first order differential equation section in 2nd semester calculus.
 

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