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Integral proofing

  1. Nov 3, 2012 #1
    1. The problem statement, all variables and given/known data

    let f(x,t)=xe^(-xt).show that the integral I(x)=∫f(x,t)dt (integration from 0 to infinite)exists for all x>=0 . is x->I(x) continuous on [0,infinite)



    3. The attempt at a solution
    what should i use here to prove the integral exist ???once i prove that exist, can i use the specific integration to see its continuity?????
     
  2. jcsd
  3. Nov 3, 2012 #2

    jbunniii

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    For a fixed value of [itex]x[/itex], are you able to show that the function [itex]g:\mathbb{R}\rightarrow\mathbb{R}[/itex] defined by [itex]g(t) = xe^{-xt}[/itex] is continuous for all [itex]t[/itex]? If so, then it can be integrated on any finite interval, so [itex]\int_{0}^{T} g(t) dt[/itex] exists and is finite for all [itex]T > 0[/itex]. You should be able to evaluate this integral explicitly to get some function [itex]G(x,T)[/itex]. Then check whether
    [tex]\lim_{T \rightarrow 0}G(x,T)[/tex]
    exists.
     
  4. Nov 3, 2012 #3
    yes, i have solved it out the limit is 1 , then means I(x)>0 but i feel confused about the continuous part,what should i do in the secound part?
     
  5. Nov 3, 2012 #4

    Dick

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    Ok, so I(x)=1 if x>0. What's I(0)?
     
  6. Nov 4, 2012 #5
    ok, i get it , thanks so much
     
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