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What is a proper way to change the differential of an integral? For example suppose we have the following integral:

$$\int_1^2 2x dx$$

which equals 3.

But we know that $2x dx = d(x^2)$. Should I write:

$$\int_1^2 2x dx= \int_1^2 d(x^2)$$

or

$$\int_1^2 2x\cdot dx= \int_1^4 d(x^2)$$

This is how I would interpret and calculate the integrals. With the first notation:

$$\int_1^2 d(x^2)=x^2\Big|_1^2=3$$

whereas with the second notation:

$$\int_1^4 d(x^2)=x^2\Big|_1^4=16-1=15$$

In essence my problem is how I should interpret the limits and the differential inside the integral.

I picked just this example but I can generalize my confusion to any integral with arbritrary limits:

$$\int_a^b f(x)dx = \int_a^bdF$$

or

$$\int_a^b f(x)dx = \int_{F(a)}^{F(b)}dF$$

where $dF=f(x)dx$ that is $f(x)$ is the derivative of $F(x)$.

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    $\begingroup$ It really depends on how you continue from there, can you complete each calculation so we can see what you have in mind? $\endgroup$ May 4, 2021 at 21:09
  • $\begingroup$ @hamam_Abdallah Could I also be more correct by denoting $dF$ as $dF(x)$? $\endgroup$
    – Anton
    May 4, 2021 at 21:36
  • $\begingroup$ IMHO as long as you know what you are talking about, it is completely irrelevant to bother with notation unless of course you are talking to someone else :-) $\endgroup$ May 4, 2021 at 21:53
  • $\begingroup$ @Buraian I agree with you but I was just wondering what is the proper way that clears all the confusions. $\endgroup$
    – Anton
    May 4, 2021 at 22:01

3 Answers 3

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I think the confusion is in what the limits refer to, lets look at your example: $$\int\limits_a^b2x\,dx=\int\limits_{x=a}^{x=b}2x\,dx=\left[x^2\right]_{x=a}^{x=b}=b^2-a^2$$ whereas: $$\int_{a^2}^{b^2}d(x^2)=\int\limits_{x^2=a^2}^{x^2=b^2}d(x^2)=\left[x^2\right]_{x^2=a^2}^{x^2=b^2}=b^2-a^2$$

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    $\begingroup$ You nailed it. With this notation no one would have a confusion. I am accepting this as an answer. $\endgroup$
    – Anton
    May 6, 2021 at 17:06
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If dF= f(x)dx then $\int_a^b f(x)dx= \int_a^b dF= F(b)-F(a)$. On the other hand, $\int_{F(a)}^{F(b)} f(x)dx$ would be F(F(b))- F(F(a)).

The first is what you want.

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  • $\begingroup$ What bothers me is that maybe someone can abuse the notation and write $\int_a^b dF= b-a$. Would it be better if I write either $$\int_a^b dF(x)$$ or $$\int_{F(a)}^{F(b)} dF$$? $\endgroup$
    – Anton
    May 5, 2021 at 11:59
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You may write $$\int_a^b f(x)dx = \int_{x=a}^{x=b} dF$$ I think this would be more accurate

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    $\begingroup$ As a professor responded to my saying, "by abuse of notation", "Let's not be that abusive!" $\endgroup$
    – user247327
    May 5, 2021 at 14:44

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