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More general formula for integrals

  1. Feb 9, 2013 #1
    I was wondering: Is there an even more general formula for the integral than int(x^k) = (x^(k+1))/(k+1) that accounts for special cases like int(x^(-1)) = ln|x| and possibly u substitutions?
     
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  3. Feb 9, 2013 #2

    mfb

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    You can combine both in a single formula:
    "int(x^k) = (x^(k+1))/(k+1) for k!=-1, int(x^(-1))=ln(|x|)"
    Apart from that... no.
     
  4. Feb 9, 2013 #3

    lurflurf

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    use limits

    $$\int \! x^k \, \mathrm{d}x=\lim_{a \rightarrow k+1} \frac{x^a}{a}+\mathrm{Constant}$$

    That is a removable singularity. When we write it in terms of usual functions we appear to be dividing by zero, but we could define a new function without doing so. Other examples include
    sin(x)/x
    log(1+x)/x
    (e^x-1)/x
    (sin(tan(x))-tan(sin(x)))/x^7

    going the other way we can define the function of two variables
    $$\mathrm{f}(x,k)=\int \! x^k \, \mathrm{d}x$$
    without any worry about dividing by zero
     
  5. Feb 10, 2013 #4

    mfb

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    For k=-1, that limit is zero for x=0 (which does not fit to the ln), and it is undefined everywhere else. As simple example, consider x=1, where you get the limit of 1/a for a->0.
     
  6. Feb 10, 2013 #5

    Mute

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    There was a "-1" missing in the numerator, which I added in the quoted equation above. Note that for ##k \neq -1##, the -1/a term can be absorbed into the integration constant.
     
  7. Feb 10, 2013 #6
    This is a funny question !
    May be, more intuitive if presented on the exponential forme, such as :
     

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  8. Feb 10, 2013 #7

    mfb

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    Ah, that makes sense.
     
  9. Feb 12, 2013 #8

    Char. Limit

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    The general definition of the integral that I use is:

    [tex]\int_a^b f(x) dx = \lim_{\text{max} \Delta x_k \to 0} \sum_{k=1}^n f(x_k^*) \Delta x_k[/tex]

    Not very useful, but it's definitely general.
     
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