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I thought that the $dx$ sign can't be changed to $du$. For example, instead of $x=u$, If I had $x=2$, then I would have to evaluate $F(2)=\int f(2) \: d2 $, but this can't be right. It makes sense to define a function $F(x)=\int f(x)\:dx$ then say that $F(u)=\int f(u)\:du$?

Pinteco
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1 Answers1

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Well, let us make some clarifications over the symbol $\int f(x) dx$. In general, whith this, we are used to note a whole family of functions, not just one function - you can see the specific construction of this family in the first part of this answer. So, writing: $$F(x)=\int f(x)dx$$ has no meaning, since the left hand side of this equation is a function and the right one is a set of functions. Even writing: $$F(t)=\int f(x)dx$$ has no meaning, for the same reasons.

So, keeping in mind that the indefinite integral is a symbol for a set of functions - with a common property - what one could use is the symbol of the definite integral, as mentioned in the comments section by @Mark Viola. So, for instance, one can write: $$F(x)=\int_a^xf(t)dt$$ which is, if $f(x)\geq0$ over its doamin, the area under $f$ from $a$ to $x$ - or the opposite of this area, if $x<a$. Note that "inside" the integral we use a different variable than that we use at its limits. So, one can also write: $$F(x)=\int_a^xf(u)du$$ Using the same variable "inside" and "outside" the integral would cause a large confusion as of with respect to what are we integrating.

You can also make both limits of integration be variable: $$F(x)=\int_x^{x+1}f(t)dt$$ or, more generally, if $g,h$ are two "nice" functions - for instance, continuous - we can consider: $$F(x)=\int_{g(x)}^{h(x)}f(t)dt$$

Note, however, that the following has a meaning: $$F(x)=\int_a^xf(x)dt$$ To interpret this, one can think as follows: $dt$ tells us with respect to which variable are we integrating. So, in our case, this variable is $t$ and, hence, everything that does not contain $t$ is considered to be a fixed number. So, for this case $f(x)$ is a plain number, and, we have: $$F(x)=\int_a^xf(x)dt=f(x)\int_a^x1dt=f(x)(x-a)$$ So, this was a more "tricky" way to write the function $f(x)(x-a)$.

Finally, we can have even more complex cases, as the following: $$F(x)=\int_a^xg(x-t)f(t)dt$$ but the discussion has already gone too far...

Vassilis Markos
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  • Can I say that $F(x)=\int_a^x f(t)dt = \int_a^x f(u)du $ ? – Pinteco Jan 04 '18 at 00:20
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    Oh, yes, in this case, the variable inside the interal can have any "name" as long as the change is applied to all the appearances of it inside the integral, with $dt$ included. Just avoid using the same name inside an outside the integral when this variable is also included in $dt$ symbol. – Vassilis Markos Jan 04 '18 at 00:26
  • What about the indefinite integrals: $\int f(t)dt=\int f(u)du$ ? – Pinteco Jan 04 '18 at 01:13
  • I'm trying to prove a theorem, that states that if $\int f(t)dt$ is non elementary than $\int f^{-1}(t)dt$ is also non elementary. But in my proof I write $\int f^{-1}(t)dt$ in function of $\int f(u) du$. If I assume that $\int f(t)dt$ is non elementary than is also $\int f(u) du$ ? I putted this question in here, and the answers appears $F(x)=\int f(x)dx$ which you said don't make sense, so i'm not conviced of the proofs that I received. – Pinteco Jan 04 '18 at 01:19
  • Can you post a link of that question. The most possible scenario is that the answerer intented to write something like $$F(x)=\int_a^xf(t)dt$$, but just used indefinite integrals due to possible time restrictions. – Vassilis Markos Jan 04 '18 at 10:53
  • Here is the orginal question: https://math.stackexchange.com/questions/2588305/proof-of-this-theorem-about-non-elementary-integrals Here is the follow up question: https://math.stackexchange.com/questions/2589588/prove-that-if-int-fx-dx-is-non-elementary-then-int-f-1x-dx-is-also – Pinteco Jan 04 '18 at 17:36
  • So, in these questions, $F(x)=\int f(x)dx$ is used with the notion "F(x)" is a function such that $F'(x)=f(x)$. As explained, this is in general wrong, but, mathematicians being the most lazy people in the world, makes many mistaken, however comfortable, pactices being used very often. – Vassilis Markos Jan 04 '18 at 17:58
  • So in these questions, if the integral $\int f(u)du$ appears, I can call this integral $F(u)$? Even if i define first that $F(x)=\int f(x) dx$. – Pinteco Jan 04 '18 at 18:03
  • Yes, having always in mind that it is meant that this is a function that has as its derivative the function $f$. – Vassilis Markos Jan 04 '18 at 18:04