This time XCAS and hp prime are really wrong! ! !

[yangyongkang]

Sorry, I am picking up XCAS or hp prime again. The first few times I wrote is the problem that hp prime or XCAS can't solve. This time I pointed out that hp prime or XCAS is really wrong. The prototype is to calculate int(abs(sin(x)/e^x), x, 0, +infinity). We know that the integrand removes the absolute value and becomes int(sin(x)/e^x,0, +infinity), using the Newton-Leibnitz formula is very good, XCAS gives an answer of 1/2, in fact the answer is correct, but after adding the absolute value, the XCAS answer is still given 1/2. Obviously wrong. We enter on Wolfram Mathematica 11.3

Code:

Integrate[Abs[Sin[x]/E^x], {x, 0, Infinity}]

I got the answer after running,

Code:

FullSimplify[%]

Code:

1/2 Coth[\[Pi]/2]

In addition, in fact, hp prime or XCAS can give int(e^(-x^2), x), and should also give int(sin(x)/e^(x^2), x) , but unfortunately, XCAS returned to the original

[parisse]

You should have noticed in the Terminal (on the Prime, or in green in Xcas) that "the discontinuities at 0 of sin(x) were not checked". Therefore it is your responsability to check the answer.
Running int(abs(sin(x)/e^x), x) returns exp(-x)*(-cos(x)*sign(sin(x))/2-sin(x)*sign(sin(x))/2), and you can check that the antiderivative is not continuous if sin(x)=0 because the cos does not vanish.
If you run int(abs(sin(x)/e^x), x, 0,10*pi) you will get 1/2+ a sum of exponentials. It's easy to guess that the answer as 1/2+sum(exp(-k*pi),k,0,inf)=1/2+1/(exp(pi)-1)
It's not too hard to implement automatic control if the integral has finite boundaries but much harder if one boundary is infinite if you want to have a reasonable responding time for common inputs.

About int(sin(x)*exp(-x^2)), you can get it as the imaginary part of int(exp(i*x)*exp(-x^2)) but unfortunately I don't have a simple form for im(erf(x-i/2)) and then it would be difficult to handle in a limit.

By the way, it is probably not intented, but I would appreciate if you could comment in a more friendly way. I do my best to improve Xcas, but I do not have all the ressources of Wolfram and I do not necessarily have the same priorities, for example I think that Xcas multivariate polynomial computing kernel is way more efficient than Mathematica.

[yangyongkang]

(12-28-2018 06:40 PM)parisse Wrote:  You should have noticed in the Terminal (on the Prime, or in green in Xcas) that "the discontinuities at 0 of sin(x) were not checked". Therefore it is your responsability to check the answer.
Running int(abs(sin(x)/e^x), x) returns exp(-x)*(-cos(x)*sign(sin(x))/2-sin(x)*sign(sin(x))/2), and you can check that the antiderivative is not continuous if sin(x)=0 because the cos does not vanish.
If you run int(abs(sin(x)/e^x), x, 0,10*pi) you will get 1/2+ a sum of exponentials. It's easy to guess that the answer as 1/2+sum(exp(-k*pi),k,0,inf)=1/2+1/(exp(pi)-1)
It's not too hard to implement automatic control if the integral has finite boundaries but much harder if one boundary is infinite if you want to have a reasonable responding time for common inputs.

About int(sin(x)*exp(-x^2)), you can get it as the imaginary part of int(exp(i*x)*exp(-x^2)) but unfortunately I don't have a simple form for im(erf(x-i/2)) and then it would be difficult to handle in a limit.

By the way, it is probably not intented, but I would appreciate if you could comment in a more friendly way. I do my best to improve Xcas, but I do not have all the ressources of Wolfram and I do not necessarily have the same priorities, for example I think that Xcas multivariate polynomial computing kernel is way more efficient than Mathematica.

I did not compare the meaning of XCAS and Wolfram Mathematica. I think the reason why it is misunderstood is that my English is too bad. I sometimes have to use Google translation, which leads to a misunderstanding of semantics. China attaches great importance to the discipline of English, but the society lacks the atmosphere to learn English. Moreover, the differences between Chinese and Western cultures are relatively large, which leads to some misunderstandings in language communication. This is inevitable and requires everyone to be tolerant. I am not malicious, I really like hp prime and XCAS, I just raised some questions. Soory my poor english