Substitution in integral

[Jiacheng Ge]

dt means with respect to t.

dw means with respect to w.

w’(t)dt means take derivative to w with respect to t.

How can we substitute w’(t)dt with dw?

dw is a prepositional phrase but w’(t)dt is a complete action.

[Nan Choi]

This is a really basic way to look at it:

Since $w'(t)$ means the derivative of $w$ with respect to $t$, we can write:

$w'(t) = \frac{dw}{dt}$

So, multiplying both sides by $dt$ gives:

$w'(t)dt = dw$

[Jiacheng Ge]

dt cannot exist and appear alone. How can we multiply dt on both sides since dt is not a number?

[Tzu-Ching Yen]

dt can exist alone. Look up the technique called separation of variable.

[Victor Ivrii]

dt cannot exist and appear alone. How can we multiply dt on both sides since dt is not a number?

Please reread "Differential" in Calculus I

[Jiacheng Ge]

After reading it, I have a follow-up question. What's the difference between  w’(t)dt,  dw,  dw/dt,  dw(t)/dt?

[Victor Ivrii]

After reading it, I have a follow-up question. What's the difference between  w’(t)dt,  dw,  dw/dt,  dw(t)/dt?

$w’(t)dt=dw$ and $dw/dt=dw(t)/dt$. I am afraid, with questions like this,  you need to retake Calculus I.

[Omar Kidwai]

Victor, I would note that such a thing is not rigorously defined, at least not in Calculus I...

I like to think of $dt$ and $dw$ written alone (as opposed to, e.g. $dw/dt$) as semi-meaningless symbols which you manipulate as a mnemonic for actual meaningful operations. You can do everything without them if you like.

[Victor Ivrii]

Omar, indeed, one can do without $dw$ etc but it would require too much efforts. Too much pain with a little gain.