Integration Question: Solutions & Tips

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Discussion Overview

The discussion revolves around the integration of the function 1/x, particularly focusing on the behavior of the integral as the limits approach zero and the implications of integrating functions that approach infinity. Participants explore the conceptual challenges posed by integrating near singularities and the nature of areas under curves that do not intersect the y-axis.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant states that the integral of 1/x results in ln|x| + C, and notes that integrating from a to b leads to ln|b| - ln|a|, which diverges as a approaches 0.
  • Another participant acknowledges a misuse of notation in their initial question and reflects on their thoughts prompted by a paper discussing concepts of zero and infinity.
  • A participant questions the nature of the area between the curve of 1/x and the y-axis, suggesting that while this area diminishes, it never fully resolves due to the curve approaching infinity.
  • There is a parallel drawn to the function 1/x², where the definite integral is said to have a value, provided the lower limit is not at 0.

Areas of Agreement / Disagreement

Participants express differing views on the implications of integrating functions that approach singularities. There is no consensus on the interpretation of areas under these curves or the handling of limits approaching zero.

Contextual Notes

Participants highlight limitations in their discussions regarding the handling of limits and the definitions of integrals near singularities, but these remain unresolved.

KingBigness
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The integral of 1/x with respect to x is ln|x| + C, so if you took the integral from a to b, you would get
ln|b|-ln|a|
and as a goes to 0, this would be infinity.

(Note that you shouldn't say you integrate 1/x from 0 to x. You can't have the limit be the variable you are integrating. If you think about what that would mean, you will see that the integral 0f f(x) from 0 to x is impossible to make sense of, since x would vary from 0 to x, whatever that means)
 
Yer I did abuse the notation a little, sorry for that.

And sorry for the question as soon as I posted it I realized how stupid it was, hence the delete.

I was just reading a paper on zero and infinity and it got me thinking...the paper didn't really like infinity.
 
What I was questioning is, because the curve never reaches the y-axis (x=0), except for at infinity, then the little area between the curve and the y axis, whilst getting smaller and smaller never quite ends.
 
KingBigness said:
What I was questioning is, because the curve never reaches the y-axis (x=0), except for at infinity, then the little area between the curve and the y axis, whilst getting smaller and smaller never quite ends.

The same could be said for 1/x2, but the definite integral does have a value (assuming the lower limit isn't at 0).
 

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