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Function f(x)=1/x intergral

  1. Jan 25, 2005 #1

    Aki

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    I have the function f(x)=1/x , and there's two asymptotes, the x-and y-axes. As x gets larger, the f(x) becomes smaller, but it is never 0. So my question is, what is the intergral from x=1 to x=infinite?
     
  2. jcsd
  3. Jan 25, 2005 #2
    [tex]\int \frac{1}{u}\,du=\ln{|u|}+C[/tex]
     
  4. Jan 25, 2005 #3
    The derivative of ln x is 1/x so like the person stated above

    [tex]\int (1/u)du = ln|u| + C[/tex]
     
  5. Jan 25, 2005 #4

    Aki

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    I'm sorry, but I'm so lost already. Could somebody please explain it to me? What is u?
     
  6. Jan 25, 2005 #5
    Let me put it in a simpler form, just replace u with x something you are probably more common to seeing.

    Now the derivative for the ln x = 1/x.

    now if you have [tex]\int {1/x}dx[/tex] you know that's the derivative of the ln of x, so you end up with that = [tex]ln|x| + C[/tex]

    this is just based of knowing the derivative and antiderivative of ln x, thats all you need to know.
     
  7. Jan 25, 2005 #6

    kreil

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    integrals at infinity are calculated by

    [tex]\int_1^{\infty}f(x)dx=\lim_{t{\rightarrow}\infty}\int_1^tf(x)dx[/tex]

    if you use this in combination with the info above you can calculate it
     
  8. Jan 25, 2005 #7

    JasonRox

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    I never learned that yet. That's pretty cool.
     
  9. Jan 28, 2005 #8

    Aki

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    so basically, there's not "number" answer to that questions? The answer is just a function?
    and also where did ln(x) come from?
     
  10. Jan 28, 2005 #9
    [tex]\int_1^{\infty} \frac{1}{x}dx[/tex] is undefined, or infinite.. depend on which one you feel more comfortable

    where did ln x came from...hmmm... it came from [itex] \frac{d}{dx} lnx = 1/x [/tex].... so your next question is why this is true.....

    assume you know product rule and the derivative of e^x is e^x itself

    [tex] e^{\ln{x}} = x[/tex]

    [tex] \frac{d}{dx} e^{\ln{x}} = \frac{d}{dx} x [/tex]

    [tex] e^{\ln{x}} \frac{d}{dx} (\ln{x}) =1[/tex] --------product rule

    [tex] x \frac{d}{dx} (\ln{x})=1[/tex]

    [tex] \frac{d}{dx} (\ln{x}) = \frac{1}{x} [/tex]

    so the anti-derivative of 1/x is ln(x)
     
  11. Jan 28, 2005 #10

    Zurtex

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    Erm vincentchan don't you mean the chain rule?
     
  12. Jan 28, 2005 #11

    dextercioby

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    Yes,it is the chain rule...Anyway,the result is correct and the method of finding it is correct as well...

    Daniel.
     
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