Thomas Calculus 5.6.15 exercise gets "hypergeom"

sigh

Comment on 2008-08-24 13:15 from acer ( Maple Rating 4

1377)

I too got the hypergeom form from Maple 12 (and 11.02, and 10.02). Which is a pity.

> In := Int(sqrt(t^5+6*t)*(5*t^4+6), t=0..1):

> value(In);
10  1/2
-- 6    hypergeom([-1/2, 11/8], [19/8], -1/6)
11
 
          1/2
     + 4 6    hypergeom([-1/2, 3/8], [11/8], -1/6)

The example (from a textbook it seems) is of course easy to do by hand. And it can be forced from Maple as well, by doing something as simple as the obvious change of variables.

> student[changevar](u=t^5+6*t,In,u);
                                    7
                                   /
                                  |    1/2
                                  |   u    du
                                  |
                                 /
                                   0

> value(%);
                                        1/2
                                    14 7
                                    -------
                                       3

Of course, as delta7 noted, the indefinite integration result could also be used. And for fun,

> sol := value(In);
       10  1/2
sol := -- 6    hypergeom([-1/2, 11/8], [19/8], -1/6)
       11
 
          1/2
     + 4 6    hypergeom([-1/2, 3/8], [11/8], -1/6)

> sol2 := value(simplify(combine(convert(convert(sol,MeijerG),Sum))));
                 54  1/2                       169    81
         sol2 := -- 6    hypergeom([-1/2, 3/8, ---], [--, 19/8], -1/6)
                 11                            88     88

Now, how can one get back to 13/3*sqrt(7) from that? Should we try to look them up? Better still if `int` could use a simpler method (such as easy substitutions) and get the simpler answer directly.

acer

sigh^2

Comment on 2008-08-24 14:04 from Axel Vogt ( Maple Rating 4

955)

Using an arbitrary upper bound and specializing later works:

  v:=int(sqrt(t^5+6*t)*(5*t^4+6), t=0..b):
  eval(v,b=1);
                                   1/2
                               14 7
                               -------
                                  3

I*sigh^2

Comment on 2008-08-24 14:37 from Alex Smith ( Maple Rating 3

294)

But look at this!

A:=int(sqrt(t^5+6*t)*(5*t^4+6),t=0..I) ;
#the upper limit is I=sqrt(-1) in case the font makes this unclear

B:=int(sqrt(t^5+6*t)*(5*t^4+6),t=I..1) ;

simplify(A+B);

This behavior of int returning a hypergeom that is difficult to simply to 14*sqrt(7)/3 is one of things that can make using MapleTA in calculus II frustrating.

It's hard to get Maple to recognize that the hypegeom expression is equal to 14*sqrt(7)/3.

Since Maple 9

Comment on 2008-08-24 15:28 from jakubi ( Maple Rating 4

923)

Maple 9.03 outputs these hypergeometrics. Up to Maple 8 the output was 14/3*sqrt(7).

Manual limits without change of variables

Comment on 2008-08-24 15:19 from gkokovidis ( Maple Rating 4

686)

Substituting limits after the integration seems to work just fine, but that does not explain why it can't be done in one step.

>restart:

>ans:=int(sqrt(t^5+6*t)*(5*t^4+6),t);

= 2/3*t*(t^4+6)*(t^5+6*t)^(1/2)

>eval(subs(t=1,ans)-subs(t=0,ans));

14/3*7^(1/2)

Regards,
Georgios Kokovidis
Dräger Medical

yes

Comment on 2008-08-24 17:35 from acer ( Maple Rating 4

1377)

Yes, delta7's first response made that point, about using the antiderivative.

And, yes, Maple is capable of generating the nicer answer. As was mentioned, that happened in some earlier versions of Maple.

The thing is, there are lots of ways to accomplish some definite integration problems. There is FTOC, which we see allows Maple to get a nice form. There is change of variables, which also works nicely here. Pattern matching might also get this one smoothly. (Search for MeijerG and integration and you can find lots of hits, many even on this site.)

Setting infolevel[int] to 3, and running the original definite integration call, one can see that Maple 12 uses an elliptic integral methodology.

It can be very difficult to tell in advance which methods will work best for a given problem -- which will both succeed and get the nicest answer, fastest. The front dispatcher of `int` has to do this. And it must be an art form, to have it please most of the people most of the time. I could envision that tweaking it to "fix" one particular example could likely "break" many other examples.

Maybe this state of affairs could be improved by crude use of Threads. Run several methods in parallel, and take which ever solution is nicest, to return to the user.

acer

correction

Comment on 2008-08-25 15:21 from acer ( Maple Rating 4

1377)

Ignore what I wrote about an elliptic routine being used. I was fooled by the userinfo messages, and should have given it more thought. The form of the result could be taken as a hint. Following IntegrationTools:-Definite:-Main one can see that the solution is obtained from`int/definite/meijerg` .

acer

The reason ...

Comment on 2008-08-24 15:35 from edgar ( Maple Rating 2

181)

I think here is the reason for the answer ... the "trick" substitution that works in the original problem does not work on either of the two terms when it is expanded...

fexpanded := expand(sqrt(t^5+6*t)*(5*t^4+6));

map(int,fexpanded,t=0..1);

map(int,fexpanded,t=0..u);

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algebraic integration

Comment on 2008-08-25 02:07 from jakubi ( Maple Rating 4

923)

One difference that I observe tracing Maple V R5.1 and Maple 12 with this integral is that the latter, apparently, does not try indefinite algebraic integration. It can be done by hand:

Int(sqrt(t^5+6*t)*(5*t^4+6),t=0..1):
subs(_X=t,`int/algebraic2/algebraic`(subs(t=_X,op(1,%))));


                               5       3/2
                           2 (t  + 6 t)
                           ---------------
                                  3

And then evaluate at the limits. I wonder if there is something wrong with this subroutine. Ie why it is not used?

With so many integration methods to choose, an option to tell: "try this one" could be more useful than expect a fully automated algorithm to catch it.