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Undefined Integral

  1. Nov 24, 2012 #1
    1. The problem statement, all variables and given/known data

    Solve the following integral:

    ∫arctg(1/√x)dx


    2. Relevant equations

    ∫u'/1+u^2 = arctg(u)+c
    ∫u.v' = u.v - ∫ u'.v

    3. The attempt at a solution

    So , I tried to define u as arctg(1/√x) but I'm having trouble finding the du. The derivate of arctg(1/√x) is -1/2*x^(3/2), so it stays -1/2*x^(3/2)/1-(1/√x)^2 (converting to the derivate u'/1+u^2) ? I was thinking about getting the derivate of arctg(1/√x) and doing this by parts. dv = dx , u = arctg(1/√x)
     
  2. jcsd
  3. Nov 24, 2012 #2

    LCKurtz

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    The derivative of ##\arctan(x^{-\frac 1 2})## is$$
    \frac 1 {1+\frac 1 x}\left(-\frac 1 2 x^{-\frac 3 2}\right )=
    -\frac 1 2 \frac{x^{-\frac 1 2}}{x+1}$$That might help your integration by parts.
     
  4. Nov 24, 2012 #3
    Thank you, now I got arctg(1/√x)*x - ∫((√x)/2x+2)*x . Do I need to make the integral by parts again?
     
  5. Nov 24, 2012 #4

    Ray Vickson

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    You should not get u'/1+u^2, which reads as
    [tex] \frac{u'}{1} + u^2.[/tex] You should have gotten u'/(1+u^2), which reads as
    [tex] \frac{u'}{1+u^2}.[/tex] Parentheses are important!

    RGV
     
  6. Nov 24, 2012 #5
    yes, I got this x.arctg(1/√x)+∫x/sqrt(x)*(2x+2) but now I'm having trouble on solving the integral :p any help?
     
  7. Nov 24, 2012 #6

    haruspex

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    Substitute for √x and use partial fractions.
     
  8. Nov 25, 2012 #7
    I don't remeber how to use the partial fractions , it's that thing with A/(x-1) * B/(x+1) and A*(x+1)*B(x-1) thing? Can't remember quite well
     
  9. Nov 25, 2012 #8

    haruspex

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    It's that thing where you have one polynomial divided by another, P(x)/Q(x). You rewrite it as a sum of simpler fractions. http://en.wikipedia.org/wiki/Partial_fractions_in_integration
     
  10. Nov 25, 2012 #9
    I transformed x/sqrt(x)*(2x+2) into sqrt(x)/2x+2 . Then I passed 2x+2 up and got Int (sqrt(x)*(2x+2)^-1) and multiplied and got Int (1/2sqrt(x)) + Int (sqrt(x)/2) but the second one is incorrect I think... Can't figure out why, tho.

    Thanks in advance
     
  11. Nov 25, 2012 #10

    haruspex

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    As I indicated, you first need to get rid of the surd by substituting a different variable for √x.
     
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