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Integral Substitutions and Mathematica

  1. Mar 5, 2013 #1
    Hi all. My professor gave us some integrals that Mathematica can't do, and we have to teach Mathematica how to do them. I got the first two, but I'm stuck with the u substitutions for these six. I know that I am supposed to make an attempt at a solution, but I've tried several different u substitutions to no avail. Sorry for the formatting.
    Any help is appreciated

    3.. ∫(x sin x ln(x sin x))/(1-Sqrt[1-Sqrt[x sin x]]) (x cos x + sin x)\[DifferentialD]x

    4. ∫(cos x -x sin x) Sqrt[x cos x] Sqrt[1+x^3 cos^3 x]\[DifferentialD]x

    5. ∫((1+ln x) Sqrt[1+x ln x])/Sqrt[x ln x] \[DifferentialD]x

    6. ∫(1-2/x^3)Sqrt[1-(x+1/x^2)^2]\[DifferentialD]x

    7. ∫(1+ln x)Sqrt[1-x^2(ln x)^2]\[DifferentialD]x

    8. ∫x^x Sqrt[x ln x](1+ln x)\[DifferentialD]x

    Note: My professor said that #8 will include a function called Erfi[x]. I have no idea what that is.

    Again, any help is appreciated.
     
  2. jcsd
  3. Mar 6, 2013 #2

    SteamKing

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  4. Mar 6, 2013 #3

    pasmith

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    The combination [itex]x \sin x[/itex] occurs frequently here as an argument to other functions and, by fortune or design, the derivative of [itex]x \sin x[/itex] is [itex]x \cos x + \sin x[/itex].

    This suggests [itex]u = x \sin x[/itex] as a first substitution, although further substitutions may be necessary.

    Similar first substitutions suggest themselves for the others, although further substitutions might be necessary.
     
  5. Mar 6, 2013 #4
    For #3, I've reduced it, with some algebra, to [itex]\frac{1}{4i}\int\frac{(xe^{ix}-xe^{-ix})(ln(x)+ln(e^{ix}-e^{-ix})-ln(2i))((x-i)e^{ix}+(x+i)e^{-ix})}{1-\sqrt{1-(\frac{1}{2}-\frac{i}{2})\sqrt{xe^{ix}-xe^{-ix}}}}dx[/itex]. However, it looks a little...complex. [/lolsofunnymathpunsftw]

    I think a u-sub of some complex exponential might be good, but I can't be sure until I try.
     
  6. Mar 6, 2013 #5
    Thanks for all the help so far. I really appreciate it. This isn't due until Friday in U.S time, but I will try some of your suggestions.
     
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