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Trouble with a line in Goldstein

  1. Jan 9, 2012 #1
    Hey, I'm looking through Goldstein's and I'm looking at equation 3.51 where it basically says

    [tex]\int \frac{dx}{\sqrt{\gamma x^2 + \beta x + \alpha}} = \frac{1}{\sqrt{-\gamma} } arccos \left( - \frac{\beta + 2 \gamma x}{\sqrt{\beta^2 - 4 \gamma \alpha} }\right) [/tex]

    Every integral book I look at says it should be what he gets except an arcsin function and the argument is positive not negative. but

    [tex] arcsin(-x) \neq arccos(x) [/tex]

    What am I missing?
  2. jcsd
  3. Jan 9, 2012 #2

    D H

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    You are missing [itex]\pi/2[/itex], which can be absorbed into the (unspecified) constant of integration.

    Note that
    [tex]\frac{d}{dx}\arccos(-x) = \frac{d}{dx}\arcsin(x) = \frac 1{\sqrt{1-x^2}}[/tex]

    and that
    [tex]\arccos(-x) - \arcsin(x) = \frac{\pi}2[/tex]
    Last edited: Jan 9, 2012
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