Is it possible to do an integral of f(x)*x without knowing f(x)?
If you know ##f(x)##, it is sometimes not even possible to express ##\int f(x) dx## using standard functions so you can expect this is also the case for ##xf(x)##, certainly when you don't know ##f(x)##!
Using integration by parts and letting u = x and dv = f(x)dx, we get
∫xf(x)dx = ∫udv = uv - ∫vdu = x∫f(x)dx - ∫f(x)dx
which is probably the most that can be said about the matter.
I am afraid that you've made a mistake.
$$ ∫vdu ≠ ∫f(x)dx $$
you defined dv = f(x)dx, so it should be v = ∫ f(x)dx
Thank you for the correction! I should have written:
∫xf(w)dw = ∫xudv = uv]x - ∫xvdu = x∫xf(w)dw - ∫x(∫zf(w)dw)dz,
or maybe I should have just left it at
∫f(w)dw = uv - ∫vdu
and kept things simple.
What was wrong with ##x## as the variable in the first place?
"What was wrong with x as the variable in the first place?"
To be technically correct, it's best to express an integral that ends up as being a function of (say) x in terms of integrating some variable that is not x (it doesn't matter which one). That's called a "dummy variable".
This is just like writing a summation in terms of an arbitrary variable whose choice does not matter:
Σ K = 15
could have been written with L or M or N, for example, in place of both instances of K.
If the integral is indefinite and ends up to be a function of (say) x, then it needs an x, of course, and the proper place for the x is as a constant of integration. (Even though after the integral is taken, x need not be thought of as a constant.)
I hope that was sufficiently confusing (:-)>.
Apart from a bit in the middle, that post is nonsense.
Hi, if it is possible to say something on ##f##, as some restriction on particular functional space or if ##f## has particular properties, then sometimes it is possible to say something also for ##\int f(x)x dx##... in other cases it is the same to consider ##\int f(x) dx## as the integration by parts shows...
"Apart from a bit in the middle, that post is nonsense."
You are correct; I wrote "constant of integration" where I meant to say "limit of integration". I hope #7 no longer seems quite so nonsensical with that correction.
I think you mean a terminal, not a limit.
One correct term for the a or b in "the integral of f(x) from a to b, with respect to x" is "limit of integration". (a is the lower limit of integration; b is the upper limit.) There may be other words for the same thing that I am not aware of.
Separate names with a comma.