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Integration of 1 / [sqrt(f(x))+g(x)] ?

  1. May 26, 2015 #1
    Hi everyone, today I came across a problem at work that requires integrating this function (indefinitely, with respect to c):

    [itex]\frac {1} {\sqrt {(k+p-c)^2 + 4 k c}-k-p+c}[/itex]

    k, p and c are all real and positive.

    I submitted it to Maxima, but it stayed implicit.

    Can you please suggest any substitution or other technique to solve it?

    Thanks
    L
     
  2. jcsd
  3. May 26, 2015 #2

    micromass

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    First rationalize the denominator: so try to bring the square root to the numerator by multiplying with ##\sqrt{(k+p-c)^2 + 4kc} + k + p - c##. Then an appropriate trigonometric substitution (like ##x = \tan(u)## or similar) will help.
     
  4. May 27, 2015 #3
    Thank you micromass, it worked!
    I multiplied both numerator and denominator by the factor you said, expanded, and got this:

    [itex]\frac {\sqrt {(k+p-c)^2 + 4 k c}+k+p-c} {4 k c}[/itex]

    which I submitted to Maxima's integrate function, and I got the result.
    Apologies for not writing it down, it's a long sum of logarithms, a square root, a linear term in c and asinh functions.
    I don't exactly see how integrating this must involve inverse hyperbolic functions, except perhaps for the known relationship:

    [itex]asinh(x)=Ln(\sqrt{1+x^2}+x)[/itex]

    Anyway, it does the job, so...
    Thanks!
    L
     
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