Understanding Integration Mechanics: Why Can't the Constant Be Taken Out?

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Discussion Overview

The discussion revolves around the mechanics of integration, specifically addressing why a constant cannot be taken out of an integral in certain cases. Participants explore the implications of this in the context of logarithmic functions and integration techniques, including applications in ordinary differential equations (ODEs).

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about why the constant cannot be factored out of the integral, providing an example involving the integral of a rational function.
  • Another participant questions the initial assumption that the constant cannot be taken out, prompting further exploration of the topic.
  • A participant presents two forms of the integral of 5x, arguing that both forms are valid and highlighting the general principle of pulling constants out of integrals.
  • There is a discussion about the equivalence of different logarithmic expressions resulting from the integration process, suggesting that constants can be represented differently.
  • A later reply indicates that the discussion has clarified the participant's understanding, particularly in relation to calculating integration factors for ODEs.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial question of whether constants can be taken out of integrals, as there are competing views and interpretations presented throughout the discussion.

Contextual Notes

Some participants reference specific forms of integrals and logarithmic identities, which may depend on the definitions and assumptions made regarding the constants involved. The discussion also touches on the application of these concepts in solving ODEs, indicating a practical context for the theoretical debate.

bp_psy
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I understand the mechanics of how this happens but i don't really understand why.

[tex]\frac{a}{b}\int\frac{1}{x+c}dx\neq\int\frac{a}{b(x+c)}dx[/tex]

Why can't the constant be taken out?:confused:
 
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bp_psy said:
Why can't the constant be taken out?:confused:

What makes you think it can't be?
 
l'Hôpital said:
What makes you think it can't be?

This:
[tex]\frac{a}{b}\int\frac{1}{x+c}dx=\frac{a}{b}ln(x+c)+C[/tex]

[tex]\int\frac{a}{b(x+c)}dx=\frac{a}{b}ln(bx+bc)+C[/tex]
 
Consider
[tex] \int 5x[/tex]

So, which is right?

[tex] \int 5x = \frac{(5x)^2}{2} + C[/tex]
or
[tex] \int 5x = 5\frac{x^2}{2} + C[/tex]
Even in regular integration, you always pull off the constants. Just because you have 1/x doesn't mean the constant shouldn't be pulled out.

However, it does worth mentioning that both your answers are actually right.

[tex] \frac{a}{b}ln(bx+bc)+C = \frac{a}{b}ln(b(x+c))+C = \frac{a}{b}ln(x+c)+ \frac{a}{b}ln b + C = \frac{a}{b}ln(x+c)+D[/tex]
Where D is just another constant.
 
l'Hôpital said:
Consider
[tex] \int 5x[/tex]

So, which is right?

[tex] \int 5x = \frac{(5x)^2}{2} + C[/tex]
or
[tex] \int 5x = 5\frac{x^2}{2} + C[/tex]
Even in regular integration, you always pull off the constants. Just because you have 1/x doesn't mean the constant shouldn't be pulled out.
I know.
l'Hôpital said:
However, it does worth mentioning that both your answers are actually right.

[tex] \frac{a}{b}ln(bx+bc)+C = \frac{a}{b}ln(b(x+c))+C = \frac{a}{b}ln(x+c)+ \frac{a}{b}ln b + C = \frac{a}{b}ln(x+c)+D[/tex]
Where D is just another constant.

Ok now I understand it. I actually run into this problem while trying to calculate integration factors for ODEs. This should simplify my calculations.

Thank you.
 

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