Why Transform Integrals of Differential Functions?

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Discussion Overview

The discussion revolves around the transformation of integrals of differential functions, particularly focusing on the integral of the natural logarithm function. Participants explore the reasoning behind transforming integrals, the application of integration techniques, and the conditions under which these transformations are valid.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the necessity of transforming the integral of a differential function into a specific form, citing examples of transcendental versus algebraic functions.
  • Another participant explains that the integral involves a differential operator and suggests using integration by parts for solving the integral of ln(x).
  • A participant reflects on the relationship between the differential of x and the differential of u in the context of substitution, leading to a reconsideration of their earlier statements about the equality of certain integrals.
  • There is a suggestion that the integral of ln(2x) can be expressed in terms of a simpler integral, but this is met with skepticism regarding the correctness of the approach.
  • One participant attempts to provide a solution without using integration by parts, but this is challenged by others who assert that integration by parts is necessary.
  • Participants express uncertainty about the correctness of their claims and acknowledge mistakes in their reasoning throughout the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correctness of the proposed methods for solving the integrals. There is a mix of agreement on the need for integration by parts, but also confusion and disagreement about the transformations and substitutions discussed.

Contextual Notes

Some participants express uncertainty about their mathematical background, which may affect their understanding of the concepts being discussed. There are unresolved questions about the validity of certain integral transformations and the application of integration techniques.

Who May Find This Useful

This discussion may be useful for individuals interested in integral calculus, particularly those exploring the nuances of integrating logarithmic functions and the application of different integration techniques.

Dumbledore
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Hello.

Can someone please explain why I have to transform an integral of a differential function into the form Integral ( lnx 1/x dx ) for example, for Integral ( lnx ).

It seems to only be done with transcendental functions and not the algebraic ones... ie. Integral ( x^2 ) != Integral ( x^2 2x dx)

Whereas, Integral (ln x) == Integral ( ln x 1/x dx)
 
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Well, the idea of an integral is that there will always be a requirement for the differential operator as it is essentially a summation of infinitely small width rectangular areas.

Taking the integral of (Ln(x)) is basically just Integral ( Ln(x) dx) and to solve that one would use integration by parts. I'm not sure where you're getting that Integral (ln x) == Integral ( ln x 1/x dx) term.
 
I think I get it, its because the differential of x is dx, and the differential of u is du. So if you have a function of a function you have to identify u and find du.

In ln x
u = x
du = dx

So in this case I am incorrect to say Integral (ln x) == Integral ( ln x 1/x dx)

But if it were ln (2x) then it would be Integral (ln (2x)) == Integral (ln(2x) 2dx)

Correct?
 
Mmm...well.

The integral of ln (2x) is simply just Integral (Ln(2x) dx). Simple as that.And then for actually solving this integral you would need to integrate by parts.
Are you trying to apply the u substitution with your statements of u = x or something?
 
And then for actually solving this integral you would need to integrate by parts.

I'm pretty sure that is incorrect, but I don't have the math background to know for sure. I see that I did make yet another mistake though... I'll show you how you can solve this without integrating by parts:

Integral( ln(2x) ) = 1/2 Integral ( ln(2x) 2dx) = 1/2 (1/2x) (2) = 1/2x

Is this not correct?
 
Dumbledore said:
I'm pretty sure that is incorrect, but I don't have the math background to know for sure. I see that I did make yet another mistake though... I'll show you how you can solve this without integrating by parts:

Integral( ln(2x) ) = 1/2 Integral ( ln(2x) 2dx) = 1/2 (1/2x) (2) = 1/2x

Is this not correct?



Sorry, but that is not correct. You need to do integration by parts.


The integral of (ln(2x)) = x*ln(2x) - x . You can differentiate it again to see that it equals ln(2x).

If you differentiate 1/(2x) that does not get you ln(2x), it just goes to some x^-2 term.
 
Yeah you are right. Basically, everything I said is completely incorrect. This entire thread is an embarrassment.
 
Dumbledore said:
Yeah you are right. Basically, everything I said is completely incorrect. This entire thread is an embarrassment.



Nah, it's all good, that's why we have these forums! Do you still have any sort of misunderstandings or confusions about this particular question?
 

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