Why Transform Integrals of Differential Functions?

In summary, the conversation discusses the process of transforming an integral of a differential function into the form of Integral (lnx 1/x dx) and the use of transcendental functions versus algebraic ones in this process. The concept of an integral and the use of integration by parts are also mentioned. There is some confusion about the correct way to solve for the integral of ln(2x), but it is eventually clarified.
  • #1
Dumbledore
33
0
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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  • #2
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.
 
  • #3
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?
 
  • #4
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?
 
  • #5
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?
 
  • #6
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.
 
  • #7
Yeah you are right. Basically, everything I said is completely incorrect. This entire thread is an embarrassment.
 
  • #8
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?
 

FAQ: Why Transform Integrals of Differential Functions?

What is a differential within an integral?

A differential within an integral is a mathematical concept that represents a small change in a variable within an integral. It is often denoted by dx, dy, or dz and is used to indicate the variable with respect to which the integral is being taken.

Why are differentials used in integrals?

Differentials are used in integrals because they allow us to account for the infinitesimal changes in a variable within an integral. This makes it easier to solve complex integrals and accurately calculate the area under a curve.

How are differentials related to derivatives?

Differentials and derivatives are closely related as they both involve small changes in a variable. Differentials are used in integrals to represent these small changes, while derivatives are used to find the rate of change of a function at a specific point.

Can differentials be used in both definite and indefinite integrals?

Yes, differentials can be used in both definite and indefinite integrals. In definite integrals, differentials are used to represent the variable of integration and in indefinite integrals, they are used to indicate the variable with respect to which the integration is performed.

Are differentials always written as dx, dy, or dz?

No, differentials can also be written as du, dv, or dw, depending on the variable being integrated with respect to. It is common practice to use differentials that match the variable being integrated, but it is not mandatory.

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