Felix E. Klee wrote:
> On Friday 25 April 2003 23:28, Raymond Toy wrote:
>
>> Felix> 2. Maxima returns a non continuous indefinite integral:
>> Felix> (C214) ratsimp(integrate(integrand,x));
>> Felix> (D214) 2*ATAN((SQRT(2)*SIN(x)+COS(x)+1)/(COS(x)+1))
>>
>>In these kinds of integrals, don't you always have to be careful about
>>what branch of atan should be used?
>>
>>This is probably why ldefint returns 0 since d214 is %pi/2 for both x
>>= 0 and x = 2*%pi.
>
>
> As far as I can see the result d214 is mathematically incorrect because:
> d214 is non contiguos at (2*n-1)*%pi.
> => d214 is underivable at (2*n-1)*%pi
> => d214 is not the antiderivative of integrand (or only the antiderivative
> on a limited range for x).
>
> A correct result would be something like
> (if (remainder(u-%pi,2*%pi) = 0) then 0 else d214)
> + 2*%pi*entier((u-%pi) / (2*%pi))
>
Perhaps so, but that goes against decades (centuries?) of mathematical
literature where such things are assumed away. :-) I don't think you'll
find any table of integrals that goes to this extreme. You're just
supposed to know. :-)
Should we also add that unknown constant of integration?
Ray