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Math Help - Fraction powered integral

  1. #1
    Newbie superphysics's Avatar
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    Fraction powered integral

    Any help solving the following integral will much appreciated:

    \int{\frac{x^{\frac{1}{2}}+x^{\frac{1}{4}}}{x^{\fr  ac{1}{3}}+x^{\frac{5}{3}}}}dx
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    Quote Originally Posted by superphysics View Post
    Any help solving the following integral will much appreciated:

    \int{\frac{x^{\frac{1}{2}}+x^{\frac{1}{4}}}{x^{\fr  ac{1}{3}}+x^{\frac{5}{3}}}}dx
    \int \frac{(x^{1/12})^6 + (x^{1/12})^3}{(x^{1/12})^4 + (x^{1/12})^{20}}dx

    Let  t = x^{1/12} \implies t' = \frac{1}{12} x^{-11/12} = \frac{1}{12} t^{-11}

    Thus, by substitution,
     12\int \frac{(t^6 + t^3)t^{11}}{t^4 + t^{20}}dt
    Simplify,
    12 \int \frac{t^{10}(t^3+1)}{t^{16}+1}dt

    But it seems the rest is going to be bad.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    \int \frac{(x^{1/12})^6 + (x^{1/12})^3}{(x^{1/12})^4 + (x^{1/12})^{20}}dx

    Let  t = x^{1/12} \implies t' = \frac{1}{12} x^{-11/12} = \frac{1}{12} t^{-11}

    Thus, by substitution,
     12\int \frac{(t^6 + t^3)t^{11}}{t^4 + t^{20}}dt
    Simplify,
    12 \int \frac{t^{10}(t^3+1)}{t^{16}+1}dt

    But it seems the rest is going to be bad.
    I got as far as that. But that's where the trouble begins. How do I get further without messing up twenty sheets of paper?
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  4. #4
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    What is so hard? It is really easy.
    Code:
           11 Pi        7 Pi
    ((-Cos[-----] - Cos[----]) 
            16           8
     
                  2           Pi
         Log[1 + x  - 2 x Cos[--]]) / 16 + 
                              16
     
             33 Pi        21 Pi
      ((-Cos[-----] - Cos[-----]) 
              16            8
     
                  2           3 Pi
         Log[1 + x  - 2 x Cos[----]]) / 16\
                               16
     
                55 Pi        35 Pi
       + ((-Cos[-----] - Cos[-----]) 
                 16            8
     
                  2           5 Pi
         Log[1 + x  - 2 x Cos[----]]) / 16\
                               16
     
                77 Pi        49 Pi
       + ((-Cos[-----] - Cos[-----]) 
                 16            8
     
                  2           7 Pi
         Log[1 + x  - 2 x Cos[----]]) / 16\
                               16
     
                99 Pi        63 Pi
       + ((-Cos[-----] - Cos[-----]) 
                 16            8
     
                  2           9 Pi
         Log[1 + x  - 2 x Cos[----]]) / 16\
                               16
     
                121 Pi        77 Pi
       + ((-Cos[------] - Cos[-----]) 
                  16            8
     
                  2           11 Pi
         Log[1 + x  - 2 x Cos[-----]]) / 16\
                               16
     
                143 Pi        91 Pi
       + ((-Cos[------] - Cos[-----]) 
                  16            8
     
                  2           13 Pi
         Log[1 + x  - 2 x Cos[-----]]) / 16\
                               16
     
                165 Pi        105 Pi
       + ((-Cos[------] - Cos[------]) 
                  16            8
     
                  2           15 Pi
         Log[1 + x  - 2 x Cos[-----]]) / 16\
                               16
     
                          Pi       Pi
       + (ArcTan[(x - Cos[--]) Csc[--]] 
                          16       16
     
              11 Pi        7 Pi
         (Sin[-----] + Sin[----])) / 8 + 
               16           8
     
                       3 Pi       3 Pi
      (ArcTan[(x - Cos[----]) Csc[----]] 
                        16         16
     
              33 Pi        21 Pi
         (Sin[-----] + Sin[-----])) / 8 + 
               16            8
     
                       5 Pi       5 Pi
      (ArcTan[(x - Cos[----]) Csc[----]] 
                        16         16
     
              55 Pi        35 Pi
         (Sin[-----] + Sin[-----])) / 8 + 
               16            8
     
                       7 Pi       7 Pi
      (ArcTan[(x - Cos[----]) Csc[----]] 
                        16         16
     
              77 Pi        49 Pi
         (Sin[-----] + Sin[-----])) / 8 + 
               16            8
     
                       9 Pi       9 Pi
      (ArcTan[(x - Cos[----]) Csc[----]] 
                        16         16
     
              99 Pi        63 Pi
         (Sin[-----] + Sin[-----])) / 8 + 
               16            8
     
                       11 Pi       11 Pi
      (ArcTan[(x - Cos[-----]) Csc[-----]] 
                        16          16
     
              121 Pi        77 Pi
         (Sin[------] + Sin[-----])) / 8 + 
                16            8
     
                       13 Pi       13 Pi
      (ArcTan[(x - Cos[-----]) Csc[-----]] 
                        16          16
     
              143 Pi        91 Pi
         (Sin[------] + Sin[-----])) / 8 + 
                16            8
     
                       15 Pi       15 Pi
      (ArcTan[(x - Cos[-----]) Csc[-----]] 
                        16          16
     
              165 Pi        105 Pi
         (Sin[------] + Sin[------])) / 8
                16            8
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  5. #5
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    Why on Earth???

    Quote Originally Posted by superphysics View Post
    Any help solving the following integral will much appreciated:
    \int{\frac{x^{\frac{1}{2}}+x^{\frac{1}{4}}}{x^{\fr  ac{1}{3}}+x^{\frac{5}{3}}}}dx
    Just passing by and was struck by the sheer absurdity of the question. Why on Earth would you want to solve (by hand!!!) an integral that has this as the answer:
    Attached Thumbnails Attached Thumbnails Fraction powered integral-integral.gif  
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  6. #6
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    Quote Originally Posted by mr fantastic View Post
    Just passing by and was struck by the sheer absurdity of the question. Why on Earth would you want to solve (by hand!!!) an integral that has this as the answer:
    Good lord.

    I had no idea it was as messy as that. I WAS making a mistake, no doubt, in the fourth step, but this bad....

    It was actually a question from a Calculus book, so I assumed it would have a neat and tidy answer, thus wasted a good hour on it.

    Thanks.
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    Quote Originally Posted by superphysics View Post
    Good lord.
    You can call me ThePerfectHacker.
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  8. #8
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    Quote Originally Posted by ThePerfectHacker View Post
    You can call me ThePerfectHacker.
    That Good Lord was a figure of speech, ThePerfectHacker!

    How'd you get that answer though? Maple or something?
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  9. #9
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    Quote Originally Posted by superphysics View Post
    How'd you get that answer though? Maple or something?
    Go to MathWorld: The Web's Most Extensive Mathematics Resource and click on Integrator.
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  10. #10
    Newbie superphysics's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post

    Oh. Thanks again.
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  11. #11
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    Quote Originally Posted by superphysics View Post
    Good lord.

    I had no idea it was as messy as that. I WAS making a mistake, no doubt, in the fourth step, but this bad....

    It was actually a question from a Calculus book, so I assumed it would have a neat and tidy answer, thus wasted a good hour on it.

    Thanks.
    And I can just hear the author(s) of that book now: "Bwahahahahaha!!".

    Out of curiosity, can you post the title and author. The Amazon review could be interesting ......
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  12. #12
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    It won't get a review. It a book by Swokowski, 4th Edition "Calculus", and a localized version at that.

    But they niggered a fish all right.
    Last edited by topsquark; December 16th 2007 at 11:35 AM.
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  13. #13
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    The Integrator sometimes may give nasty answers, but sometimes one can get a different answer and of course, not nasty.

    In this case, the integral is absolutely nasty.
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