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Integral of 1/sqrt(u^2-a^2) du, when u < -a

  1. Feb 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Show and prove that where u < -a:

    integral of 1/(sqrt(u^2-a^2)) = ln(u+sqrt(u^2-a^2))+C or -cosh^-1(-u/a)+C

    2. Relevant equations
    when u > a, the
    integral of 1/(sqrt(u^2-a^2)) = ln(u+sqrt(u^2-a^2))+C or -cosh^-1(-u/a)+C

    My attempt at a solution: Don't I just use the same methodology as if u > a?

    integral 1/sqrt(-a^2+u^2) du
    For the integrand, 1/sqrt(u^2-a^2) substitute u = a sec(s) and du = a tan(s) sec(s) ds. Then sqrt(u^2-a^2) = sqrt(a^2 sec^2(s)-a^2) = a tan(s) and s = sec^(-1)(u/a):
    = integral sec(s) ds
    The integral of sec(s) is log(tan(s)+sec(s)):
    = ln(tan(s)+sec(s))+constant
    Substitute back for s = sec^(-1)(u/a):
    = ln((u (sqrt(1-a^2/u^2)+1))/a)+constant

    = ln(sqrt(u^2-a^2)+u)+constant
     
    Last edited: Feb 11, 2012
  2. jcsd
  3. Feb 11, 2012 #2

    HallsofIvy

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    The difference is that the integral of 1/x is NOT ln(x), it is ln|x|. [itex]\int dx/x= ln|x|+ c[/itex]. Which sign you use for the absolute value will depend upon whether x> a or x< -a.
     
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