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Final Formula

  1. Dec 25, 2004 #1
    What does it mean when a question asks you to get a final formula?

    I have an example of the question that is asking for this.

    1. a) Find a reduction formula for [tex] \int(\ln x)^n dx [/tex] which I did find and got the answer: [tex]x (\ln x)^n - n \int (\ln x)^{n-1} dx [/tex].

    But question part b) asks if I can use what I have in a) to get a final formula for [tex] \int(\ln x)^n dx [/tex] ? I do not understand what do they mean by a final formula?

    Thanks for any help.
  2. jcsd
  3. Dec 25, 2004 #2
    The formula is a final formula if it has NO INTEGRALS in it. So it will just have an algebraic combination of lnx and x.
  4. Dec 25, 2004 #3


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    You're asked for a recurrence formula,which u already found.Denoting
    [tex] I_{n}=:\int (\ln x)^{n} dx [/tex]
    ,u have forund that:
    [tex] I_{n}=x(\ln x)^{n}-nI_{n-1} [/tex]
    .Make [itex] n\rightarrow n-1[/itex],and get:
    [tex] I_{n-1}=x(\ln x)^{n-1}-(n-1)I_{n-2} [/tex]
    and so on,until
    [tex] I_{0}=x(\ln x)^{0}-0I_{-1}=x [/tex]
    Use te reccurence relations to find [itex] I_{n} [/itex] as a function of "x" and "n".

  5. Dec 25, 2004 #4
    I understand this. This is reminding me of my discrete math course. From what I remember, I learned how to find an explicit formula given the recurrence relation but that was only for 1st or 2nd order, homogeneous, linear, and constant co-efficient equations. Now this is calculus and I'm confused since we have these variables and ln x. I dont know how to go about solving this. Could you show me how to solve this recurrence relation?
  6. Dec 25, 2004 #5
    Calulate [tex]I_{1}[/tex] using the first formula with [tex]n=1[/tex] and [tex]I_{0}=x[/tex]. Then calculate the next, and next, and next...until you see the pattern. Then, if required, prove your formula one way or another.
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