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Recently I was doing calculus when I looked up the integral of an expression (which I forgot) however one thing I remember about the video was that during the integration process the person split up the integral using integration by parts like so.

$$\int f(x)*g'(x) \;\mathrm{dx} = f(x)*g(x) \;\mathrm{dx} - \int f'(x)*g(x) \;\mathrm{dx}$$

They then defined $I = \int f(x)*g'(x) \;\mathrm{dx}$

and replaced the integral like so

$$I = f(x)*g(x) \;\mathrm{dx} - \int f'(x)*g(x) \;\mathrm{dx}$$

And the integral of $f'(x) * g(x)$ just so happened to be the exact same as the integral of $f(x) * g'(x)$ which allowed them to do the following.

$$2I = f(x) * g(x) $$

and then solved by dividing

$$ I = (f(x) * g(x))/2$$

My question is, to what extent is it allowed to manipulate integrals like this, can I move around integrals about an equal sign any time I want and is this also allowed when working with derivatives? Might be a stupid question but sometimes you can manipulate the dy/dx notation for derivatives, whereas sometimes you can not which is why I am not sure if moving integrals and derivatives is allowed any time.

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An integral is just a number, or a function. If two numbers / functions are equal, then they are still equal when you add something to them. So yes, this is ok. The only issue with $dy/dx$ is in the fact it looks like a fraction but isn't, really. If you treat $dy/dx$ as one object, you can never go wrong. If you treat it as a ratio of two objects, $dy$ and $dx$, then you might make an error.

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  • $\begingroup$ So I can always move integrals and derivatives, algebraicly, for example dividing by them multiplying subtracting adding and maybe even exponentiation, but unless I know what I am doing I can't split dy/dx apart? $\endgroup$
    – Dan Lupu
    Jun 25, 2023 at 16:31
  • $\begingroup$ @DanLupu Yes. I don't want you to feel like $dy/dx$ is "special" with different rules for handling it. You have to understand that $dy/dx$ is basically one symbol and that anything you do with it is ok. If you treat it like a fraction, you risk treating $dy/dx$ as a ratio of two different things, and then you can go wrong (you are no longer manipulating $dy/dx$, the derivative, you are manipulating something else which hasn't really been defined) $\endgroup$
    – FShrike
    Jun 25, 2023 at 16:39
  • $\begingroup$ So the only times that I could split up dy/dx is when I define them? like when they are defined when doing U-sub or when they are explicitly defined, for example when doing error approximation I've seen people define the differentials dy/dx as deltaY/deltaX and then moving them around to form dy = f(x)dx (is approximately equal to) $\endgroup$
    – Dan Lupu
    Jun 25, 2023 at 16:47
  • $\begingroup$ Error approximations should be treated with care. $\Delta y/\Delta x$ will always be ok to treat like a fraction since it literally is a fraction. $dy/dx$ is a limit of fractions but is not $(dy) / (dx)$ where $dy$ and $dx$ are individual numbers. $dy=f(x)dx$ is (in most contexts) an informal notation, it is not precise. In $u$-sub, when e.g. we say $u=x^2$ and $du=2x\,dx$, this is also informal notation. It is much more accurate to say $du/dx = 2x$ so $\int f(x)\cdot 2x\,dx=\int f(x)\cdot(du/dx)\,dx=\int f(\sqrt{u})\,d u$ (this is still informal notation, but it's better) $\endgroup$
    – FShrike
    Jun 25, 2023 at 17:06
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    $\begingroup$ Oh okay, this actually makes a lot more sense, thank you so much for explaining to me! $\endgroup$
    – Dan Lupu
    Jun 25, 2023 at 17:24

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