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Improper Integral Integration

  1. Feb 9, 2010 #1
    [Solved] Improper Integral Integration

    Sorry, don't know how to use the latex stuff for integrals :P

    1. The problem statement, all variables and given/known data

    Integrate the following from 0 to infinity: 1/(sqrt[x]*(1+x))

    2. Relevant equations

    Integrate 0 to 1: 1/(sqrt[x]*(1+x))
    Integrate from 1 to infinity: 1/(sqrt[x]*(1+x))

    3. The attempt at a solution

    Integrate from 1 to infinity: 1/(sqrt[x]*(1+x))
    This is convergent because you can tell as it goes to infinity, it will approach 0.

    So, I should (i think) be able to find the integral of this, but... I can't use improper fractions because that square root messes everything up (atleast, from my understanding how improper fractions work... When I try to break it up, the square root always ends up negative).

    Integrate 0 to 1: 1/(sqrt[x]*(1+x))
    This shoots off to infinity as the function approaches zero, so...

    If b = 0
    lim x -> b+ Integrate b to 1: 1/(sqrt[x]*(1+x))

    If x gets really close to zero, I can assume
    1/(sqrt[x]*(1+x)) ~ 1/(sqrt[x])

    Therefore, by using comparison tests,

    1/(sqrt[x]*(1+x)) [tex]\geq[/tex] 1/(sqrt[x])

    Since I know that (1/x^p) is convergent if p<1 for any bounds between 0-1.

    Basically, I know the theory... I just got no idea how to solve the rest of it.
    Last edited: Feb 9, 2010
  2. jcsd
  3. Feb 9, 2010 #2
    Try the substitution u = √x
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