# Fraction powered integral

• Dec 15th 2007, 08:03 PM
superphysics
Fraction powered integral
Any help solving the following integral will much appreciated:

$\int{\frac{x^{\frac{1}{2}}+x^{\frac{1}{4}}}{x^{\fr ac{1}{3}}+x^{\frac{5}{3}}}}dx$
• Dec 15th 2007, 08:11 PM
ThePerfectHacker
Quote:

Originally Posted by superphysics
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.
• Dec 15th 2007, 08:33 PM
superphysics
Quote:

Originally Posted by ThePerfectHacker
$\int \frac{(x^{1/12})^6 + (x^{1/12})^3}{(x^{1/12})^4 + (x^{1/12})^{20}}dx$

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?
• Dec 15th 2007, 08:41 PM
ThePerfectHacker
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
• Dec 15th 2007, 08:41 PM
mr fantastic
Why on Earth???
Quote:

Originally Posted by superphysics
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:
• Dec 15th 2007, 08:45 PM
superphysics
Quote:

Originally Posted by mr fantastic
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.
• Dec 15th 2007, 08:49 PM
ThePerfectHacker
Quote:

Originally Posted by superphysics
Good lord.

You can call me ThePerfectHacker.
• Dec 15th 2007, 08:54 PM
superphysics
Quote:

Originally Posted by ThePerfectHacker
You can call me ThePerfectHacker.

That Good Lord was a figure of speech, ThePerfectHacker!

How'd you get that answer though? Maple or something?
• Dec 15th 2007, 08:56 PM
ThePerfectHacker
Quote:

Originally Posted by superphysics
How'd you get that answer though? Maple or something?

Go to MathWorld: The Web's Most Extensive Mathematics Resource and click on Integrator.
• Dec 15th 2007, 08:59 PM
superphysics
Quote:

Originally Posted by ThePerfectHacker

Oh. Thanks again.
• Dec 15th 2007, 09:19 PM
mr fantastic
Quote:

Originally Posted by superphysics
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 ......
• Dec 15th 2007, 09:33 PM
superphysics
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. :)
• Dec 16th 2007, 04:31 AM
Krizalid
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.