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Decay formula with Improper Integrals

  1. Sep 27, 2008 #1
    1. The problem statement, all variables and given/known data
    Hey everyone! I have another question about improper integrals, they're so hard!

    M = -k [tex]\int[/tex][tex]\stackrel{inf.}{0}[/tex] te^(kt) dt

    When k = -0.000121 (Carbon 14's constant, we are solving for the mean life of a carbon-14 isotope)

    2. Relevant equations
    Improper integrals, integration by parts

    3. The attempt at a solution

    =- k lim[tex]_{t->inf.}[/tex] [tex]\int[/tex]te^(kt) dt from 0 to infinity

    by parts:
    u = t
    du = dt
    dv = e^(k) dt
    v = (1/k)e^(-kt)

    =(t((1/k)e^(kt)) + [tex]\int[/tex](1/k)e^(kt)dt

    =((t)/(k))e^(-0.000121t) - (1/((k)^2)e^(kt)

    =- k lim[tex]_{t->inf.}[/tex] [tex]\int[/tex](t)/(k)e^(kt) - (1/(k)^2)e^(kt)

    Where can I go from here? I can put both terms over (k)^2 but the limit of that term times e^(k) equals 1*, right? So is hte answer just -k? Somehow I am skeptical!
    Last edited: Sep 27, 2008
  2. jcsd
  3. Sep 27, 2008 #2


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    Science Advisor
    Homework Helper

    Hi lelandsthename! :smile:

    oooh … you've made this so complicated! :cry:

    why not just leave k as k until the very end?
    Nooo … e-∞ = 0, but e0 = 1. :smile:
  4. Sep 27, 2008 #3
    haha! ok I will give it a try with k first, it'll even make it look neater! :smile: and thank you for helping me with the limit! So because the limit equals one then the answer will just be -k?
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