I Integral of f(x)*x

ddddd28

Is it possible to do an integral of f(x)*x without knowing f(x)?

Math_QED

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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 $f(x)$!

zinq

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.

Chan Pok Fung

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

zinq

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.

PeroK

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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?

zinq

"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:

5
Σ K = 15
K=1​

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 (:-)>.

PeroK

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Gold Member
2018 Award
"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".

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.

Ssnow

Gold Member
Is it possible to do an integral of f(x)*x without knowing f(x)?
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...

zinq

"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.

pwsnafu

"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.

zinq

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.

"Integral of f(x)*x"

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