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

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∫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.

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I am afraid that you've made a mistake.

∫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.

$$ ∫vdu ≠ ∫f(x)dx $$

you defined dv = f(x)dx, so it should be v = ∫ f(x)dx

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∫^{x}f(w)dw = ∫^{x}udv = uv]^{x} - ∫^{x}vdu = x∫^{x}f(w)dw - ∫^{x}(∫^{z}f(w)dw)dz,

or maybe I should have just left it at

∫f(w)dw = uv - ∫vdu

and kept things simple.

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What was wrong with ##x## as the variable in the first place?

∫^{x}f(w)dw = ∫^{x}udv = uv]^{x}- ∫^{x}vdu = x∫^{x}f(w)dw - ∫^{x}(∫^{z}f(w)dw)dz,

or maybe I should have just left it at

∫f(w)dw = uv - ∫vdu

and kept things simple.

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To be technically correct,

This is just like writing a summation in terms of an arbitrary variable whose choice does not matter:

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Σ K = 15

K=1

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

I hope that was sufficiently confusing (:-)>.

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Apart from a bit in the middle, that post is nonsense."What was wrong withxas 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)xin 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, andthe 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 (:-)>.

Gold Member

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

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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.

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.

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