MaplePrimes - answers and comments on Question, Definite integral

The reasons

Alejandro Jakubi — Wed, 29 Aug 2012 08:37:06 Z

Indeed, the indefinite integral is computed:

> int(x^4/(4*x^5+2), x);
                                          5
                               1/20 ln(2 x  + 1)

and actually, the method FTOC does compute internally the value of the definite integral by taking the limits of this primitive function. But then, the discontinuity checker fails at deciding whether this function is continuous within the integration interval:

> kernelopts(opaquemodules=false):
> trace(IntegrationTools:-Definite:-Integrators:-FTOC:-HandleDisconts):
> int(x^4/(4*x^5+2), x=0..1,method=ftoc);
{--> enter HandleDisconts, args = 1/20*ln(2*x^5+1), 1/2*x^4/(2*x^5+1), x, 0, 1
, table([(UnevaluatedIntegral)=UnevaluatedInt(x^4/(4*x^5+2),x = 0 .. 1,method =
ftoc),(AllSolutions)=false,(OriginalParms)=[AllSolutions = false, 
CauchyPrincipalValue = false, continuous = false, method = ftoc],(continuous)=
false,(CPV)=false,(formula)=true,(CauchyPrincipalValue)=false,(mode)=default])
, false
                         5
disconts := {[RootOf(2 _Z  + 1, index = 1), NULL, NULL],
                5
    [RootOf(2 _Z  + 1, index = 2), NULL, NULL],
                5
    [RootOf(2 _Z  + 1, index = 3), NULL, NULL],
                5
    [RootOf(2 _Z  + 1, index = 4), NULL, NULL],
                5
    [RootOf(2 _Z  + 1, index = 5), NULL, NULL]}
<-- exit HandleDisconts (now in Apply_Main) = FAIL}
                            4
                           x
                    int(--------, x = 0 .. 1, method = ftoc)
                           5
                        4 x  + 2

The workaround is disabling this check by setting the option continuous=true:

> int(x^4/(4*x^5+2), x=0..1,continuous=true);
                                   1/20 ln(3)

So, the reasons are on the one hand the problem handling the roots of 2*x^5+1 as candidates for discontinuities of the primitive function. And on the other hand, a problem of design. As the discontinuity checker is unable to decide, the integrator hides this weakness by returning unevaluated. But how could a normal user guess this weakness and deal with it? In my opinion, it would be more useful and honest returning the result already computed with a warning message about the failure of the discontinuity checker and the need for checking this result by other means.

N.B. for 'continuous=true'

Axel Vogt — Wed, 29 Aug 2012 23:25:14 Z

Hm ...

Using continuous=true is nothing but using the anti-derivative, so it is a kind
of 'cheating' and only avoids doing that manually.

But actually Maple has the tools to find the discontinuities in that case, even
if it does not answer it for 1/20*ln(2*x^5+1):

Applying allvalues to RootOf would give it here - which is quite expensive and
would not work in general (beyond unit roots, 5 <= degree).

But "2*x^5+1; sturm(%,x,0,1);" would answer it quickly and restricting to the
case of interest - the intervall from 0 to 1.

May it would be worth to implement it for those cases, where the task is just a
polynomial problem over a real interval.

I uses Derive and got the answer ln(3)/20.

toandhsp — Thu, 30 Aug 2012 18:16:30 Z

I used Derive and got the answer ln(3)/20. We can put t = x^5.

Deep thought

Markiyan Hirnyk — Wed, 29 Aug 2012 17:21:28 Z

This works in many other cases. For example, compare

> assume(t, real); int(ln(abs(1-u^2*exp((2*I)*t)))/u^2, u = 0 .. infinity, continuous = true);

> simplify(%);
Pi |sin(t)|

with

> assume(t, real); int(ln(abs(1-u^2*exp((2*I)*t)))/u^2, u = 0 .. infinity);

1) To Alejandro Jakubi. Thanks for the

Kitonum — Thu, 30 Aug 2012 00:15:21 Z

1) To Alejandro Jakubi.

Thanks for the detailed explanation! It is unfortunate that Maple unable to cope with checking obvious statements.

Another example:

is(2*х^5 +1>0) assuming x>=0, x<=1;

false

This is just a bug!

2) To Markiyan Hirnyk.

I think that the use of continuous=true option, without proof of the convergence of the improper integral, is unacceptable because may result in an error.

An example:

int(1/x, x=-1..2, continuous=true);

-Pi*I+ln(2)

Received an incorrect result!

To force an open door

Markiyan Hirnyk — Thu, 30 Aug 2012 07:28:44 Z

@Kitonum I don't understand why you force an open door.