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Some tricky antiderivates

  1. Nov 16, 2006 #1
    I have to prepare for the exams and this is a set of integrals I can't do... some hints pls :

    1.[tex] \int \frac{x}{\sqrt{x^{2}+2x+2}} \; dx [/tex]

    2.[tex] \int \frac{x}{x^{3}+1} \; dx [/tex]

    :confused:
    Thank you for your time.
     
  2. jcsd
  3. Nov 16, 2006 #2

    siddharth

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    Hi and welcome to PF csi86!

    You need to show some work before you receive help. What are your thoughts/ideas on these problem? What have you done till now, and where are you stuck?
     
  4. Nov 16, 2006 #3

    dextercioby

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    For the second try to solve

    [tex] \frac{x}{x^{3}+1} =\frac{A}{x+1} +\frac{Bx+C}{x^{2}-x+1} [/tex]

    Daniel.
     
  5. Nov 16, 2006 #4
    For the first one use integration by parts:

    [tex] u = x [/tex]


    [tex] dv = \frac{1}{\sqrt{1+(x+1)^{2}}}\; dx[/tex]
     
    Last edited: Nov 16, 2006
  6. Nov 16, 2006 #5
    Thank you, I have succesfully solved the second one, I might have done some mistakes :
    [tex] -ln|x+1| + \frac{1}{2} ln{(x^{2}-x+1)} + \frac{2}{\sqrt{3}}\arctan{\frac{2x-1}{\sqrt{3}}} [/tex]
     
  7. Nov 16, 2006 #6

    dextercioby

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    Just don't forget the integration constant.

    Daniel.
     
  8. Nov 16, 2006 #7
    Indeed , this might hurt in an exam. :redface:

    I tried to solve the first one and I end up with :
    [tex] \frac{x^{2}}{2 \sqrt{x^{2}+2x+2}} + \frac{1}{2} \int \frac{(x^{2})(2x+2)}{2 \sqrt{(x^{2}+2x+2})^{3}}} \;dx [/tex]

    (Using integration by parts)
     
  9. Nov 16, 2006 #8

    dextercioby

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    Write it like that

    [tex] \int \frac{x}{\sqrt{x^{2}+2x+2}} \ dx =\frac{1}{2}\int \frac{d(x^{2}+2x+2)}{\sqrt{x^{2}+2x+2}} -\int \frac{dx}{\sqrt{x^{2}+2x+2}} [/tex]

    Daniel.
     
  10. Nov 16, 2006 #9
    I figured it out, thanks for the help Daniel. :)

    I end up with :
    [tex] \sqrt{x^{2}+2x+2} - \ln{(x+1+ \sqrt{x^{2}+2x+2})} [/tex]

    I don't know if that is correct but I honestly hope so :).
     
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