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I Integral of f(x)*x

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  1. Nov 8, 2016 #1
    Is it possible to do an integral of f(x)*x without knowing f(x)?
     
  2. jcsd
  3. Nov 8, 2016 #2

    Math_QED

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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)##!
     
  4. Nov 8, 2016 #3
    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.
     
  5. Nov 9, 2016 #4
    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
     
  6. Nov 9, 2016 #5
    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.
     
  7. Nov 9, 2016 #6

    PeroK

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    What was wrong with ##x## as the variable in the first place?
     
  8. Nov 9, 2016 #7
    "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 (:-)>.
     
  9. Nov 9, 2016 #8

    PeroK

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    Apart from a bit in the middle, that post is nonsense.
     
  10. Nov 12, 2016 #9

    Ssnow

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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...
     
  11. Nov 14, 2016 #10
    "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.
     
  12. Nov 16, 2016 #11

    pwsnafu

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    I think you mean a terminal, not a limit.
     
  13. Nov 16, 2016 #12
    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.
     
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