There are brainlets on Veeky Forums who think [math]C[/math] in [math]int f(x) mathrm{d}x = F(x) +C[/math] is a...

>there are brainlets on Veeky Forums who think [math]C[/math] in [math]\int f(x) \mathrm{d}x = F(x) +C[/math] is a constant

penis

>there are brainlets on Veeky Forums who would write [math] \int f(x)\mathrm{d}x [/math] instead of [math] \int f(x)\, \mathrm{d}x [/math].

but is is a constant

Not quite. It's a free variable.

then what is its value?

check mate

any real number

wrong

if it was anything other than constant then the right hand side wouldn't derive to be f(x)

>ThErE aRe BrAiNlEtS oN Veeky Forums WhO tHiNk C iN ∫f(X)dX=f(X)+c Is A cOnStAnT

>if it was anything other than constant then the right hand side wouldn't derive to be f(x)
Are you sure you understand what a free variable is?

>Are you sure you understand what a free variable is?
yes

>any real number
>constant
pick one

you can take the derivative with respect to c

which real number isn't constant?

It is though, constant refers to the type of function it is

So basically it should be written as c(x), but since its value itself is constant, the argument is left out

it's a constant in each solution of the set of solutions to the integral

but it doesn't have to assume the same value on the whole domain, it the domain is for example sum of disjoint intervals then c, or c(x) has to be constant on each of the intervals but may assume different values on different intervals, so it doesn't have to be constant but just locally constant.

>, it the domain is for example sum of disjoint intervals then c
then it's not differentiable

so tan x or 1/x^2 are not integrable?

>tan x
That is not integratable on R.

not on [math] \mathbb{R} [/math]

They are integrable on their domains, it's meaningless to talk about integral of function in points where it doesn't exist

i've always seen integrability defined in terms of a single closed interval, what definition are you using?

>They are integrable on their domains
If you look at tan:R -> R it is NOT integratable, by the only definition of integratability (the integral over the set according to the measure is finite) I have ever heard of.

tan:R -> R it is NOT a function

It's constant versus the variable of integration
Nobody in this thread got to ode apparently

Yes, just take tan:R -> R*, or ignore the infinities, they are irrelevant anyways for considering the integral.