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Integrals computation: Help me please!

  1. Oct 4, 2011 #1
    Hi all,

    can you help me to compute these integrals?

    [tex]\int \frac{x \sqrt{a x+b+x^2}}{d^2+x^2} \, dx[/tex]

    [tex]\int \frac{x}{\left(d^2+x^2\right) \sqrt{a x+b+x^2}} \, dx[/tex]

    a,b and d are real and positive.

    I tried with Mathematica, but the results involve logarithmic functions with complex argument whereas I need a solution in the real domain.

    Thank you to all in advance.
     
  2. jcsd
  3. Oct 4, 2011 #2

    Ray Vickson

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    Maple 14 delivers answers in terms of logarithms and square roots and the like. These may be real or complex, depending on the relative magnitudes of a, b and d, but they are written in a symbolic form that does not involve complex quantities explicitly. They are much too large and complicated to be reproduced here. All I can suggest is that you try a tool other than Mathematica; it is often the case that when something does not work out well using one symbolic package, it is better done in another (and there is no consisitency: no one package is universally better than another).

    RGV
     
  4. Oct 5, 2011 #3
    Thank you soo much, RGV.
    I will try with maple and I'll let you know.

    :)
     
  5. Oct 5, 2011 #4

    Ray Vickson

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    The best way to do it is to recognize that for real d and x we have Re(1/(x + I*d)) = x/(x^2 + d^2), so you can integrate g1 = sqrt(x^2 + ax + b)/(x + I*d) or g2 = 1/(x + I*d)/sqrt(x^2 + ax + b), then (by assuming a,b,x,d real) take the real part *after* doing the integrals, by using the 'evalc' command. This shorter and sweeter than a direct approach.

    RGV
     
  6. Oct 6, 2011 #5
    Thank you again RGV.

    Just one other thing. I'm using maple for the first time and I think that I'm doing something wrong because the result which I obtain is very long.
    Is this the right sequence of commands?

    [tex]int( \!{\frac { \sqrt{{x}^{2}+bx+c}}{x+id}},x) [/tex]
    [tex]assume(x, 'real', b, 'real', c, 'real', d, 'real') [/tex]
    [tex]evalc(\%) [/tex]


    Phantony.
     
  7. Oct 6, 2011 #6

    Ray Vickson

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    No, those are all wrong. You should use
    J:=int(sqrt(x^2+a*x+b)/(x+I*d),x);
    J1:=evalc(J) assuming real;
    or
    J1:=evalc(J) assuming a>0,b>0,d>0,x>0;
    If you don't want output echoed on the screen, use the command end ':' instead of ';'

    I actually prefer to use something like g1:=sqrt(x^2+a*x+b)/(x+I*d): (or ;) then put
    J:=int(g1,x). That way I can get back later to the function g1, or to the function
    f1 given as:
    evalc(g1) assuming x>0,a>0,b>0,d>0; f1:=Re(%) assuming x>0,a>0,b>0,d>0;
    This can all be shortened by setting
    assns:=x>0,a>0,b>0,d>0;
    evalc(g1) assuming assns; f1:=Re(%) assuming assns.

    RGV
     
  8. Oct 10, 2011 #7
    I have no words to express my gratitude to you.

    I solved the problem :))

    :biggrin::!!):biggrin::!!)
     
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