Adding differentials

Question

Suppose I have a sum of two Indefinite integrals, $\int f(t)dx + \int f(t)dy$. Is it possible to write this in a singke form as $\int f(t) d(x+y)$, and vise versa?

It looks “okay” to me from a logical point of view, but I obviously have no rigorous reasoning of why this should be

Edit:

I now realise that I might have stirred confusion by using a function of x. I have now changed f to a function of t. The reason I am asking is that I know that x+y=t so If i am able to express the differential in that form, integration would be possible

Answer 1

First of all, keep in mind that "d" is an operator, not a multiplier. Think of $dx$ meaning $d(x)$ and being similar to $\sin(x)$. So, if you had $f(x)\sin(x) + f(x)\sin(y)$ you could not rewrite that as $f(x)\sin(x + y)$.

That being said, as noted in the comments below, the addition rule states that $d(x + y) = d(x) + d(y)$, and this is true for both derivatives and differentials.

Therefore, let's do the requested transform a step at a time. The starting formula: $$ \int f(t)\,dx + \int f(t)\,dy $$ We can use the addition rule to combine the integrals into one: $$ \int \left(f(t)\,dx + f(t)\,dy\right) $$ Now we can associate: $$ \int f(t)(dx + dy) $$ Now we can use the addition rule in differentials to combine the differentials together: $$ \int f(t)\,d(x + y) $$ And that is the result you were looking for.

NOTE - I had originally come out against this method based on the reasoning in the first paragraph, when someone pointed me to the obvious point about differentials and addition in the second paragraph.

Answer 2

Differentiation under integral sign is possible using Leibniz Integral Rule. However, it is not the same as keeping non-operating variables constant in a sort of "partial integration" mode. And these are not differentials.

I do not exactly get why you want to do this. If you want to include integration into a generality, then it is better to convert a constant into an extra variable. This part can be ignored if it is not addressing your concern/question.

If $x +i y =z $ then $ dx + i\; dy = dz .$ Combination with single complex variable is possible by contour integration etc., provided several accompanying complex operations are also adopted.

If you want more direction flexibility with integration, there should be more arguments for functional representation itself.

If p,V are variables with a product invariant $ T= p\cdot V $ we can unfreeze T giving it status of a variable $\dfrac{pV}{T}=R $. We can see that when T is constant, it is Boyle's law; if T is roped in as a new variable, $ pV/T = R $ where R is a more general universal gas constant. We can move more freely (extra dimension) on a surface rather than scramble on a narrow line.