Integrating the simple functions 1/x, 1/x², 1/x³ upwards from 1 give correct results for low values of the upper limit but with the upper limit in the order of 200 or higher the results are not what I would expect. Please see the attached worksheet.
Is there a problem with the integral function or is there something I haven't grasped?
H
integral.sm (5 KiB) downloaded 51 time(s).

Hi. It's because the integral it's a numerical procedure, and maybe not a very robust one. So, like any numerical method, if it don't work, just use another.

integral.sm (13 KiB) downloaded 55 time(s).

Best regards.
Alvaro.

Hakelm,
I opened your file and the results were correct. Therefore I believe that the parameter for the accuracy of the integrals in your SMath installation is not correctly set.
Try to change Tools->Options->Calculation->Integral Accuracy
For example, write 1000

However, Alvaro's considerations must be kept in mind

sergio

Many thanks for two very good answers!
Now I know that Smath doesn't know even the simplest of integrals but that approximation accuracy can be improved upon.
Even if I understand my system doesn't (SMathStudioDesktop.0_99_7109.Mono on Ubuntu 16.04).
It can't find the function Rkadapt.
I guess that the Runge-Kutta functions are to be found i some plugin. Where can I find that?
H

Wrote

I guess that the Runge-Kutta functions are to be found i some plugin. Where can I find that?

ODE Solvers.

Wrote

Is there a problem with the integral function or is there something I haven't grasped?

The Simpson integrator ranges from exact to freak.
In your examples, increase the integral accuracy from
menu, options, calculations, integral accuracy [max 10000]
Wise to sanity Wolfram Alpha cost is 0.00 $

integral.sm (46 KiB) downloaded 51 time(s).

May I assume that the SMath-integral is made using Simpson's rule and that the integral accuracy [max 10000] is the number of steps taken by Simpson?
H

Wrote

May I assume that the SMath-integral is made using Simpson's rule and that the integral accuracy [max 10000] is the number of steps taken by Simpson?

Quite right: The Smath integrator is Simpson.
Ranged accuracy [50 ... 10000]. Trivial cases are exact.
Two more exact cases are know 1/x [ln(x), b_spline].
On long range of the variate 'x' the Simpson/Romberg kernel
gives a much better result than simple Simpson.
Carlos adaptive algorithm is fool proof all cases.
You want these two proposed ... reply YES/NO.
Jean

Yes please
H

As you can see, Romberg has the virtue of reducing the range of integration
in the [0 ... 1 ] domain. In two versions directly associated with the
Smath native Simpson integrator or Simpson algo style, thus integration
accuracy 'n' @ the user command line.
These two documents are like minimalist tool box.

Jean



Integrate Compendium_1 Romberg_FD_Adaptive [Carlos].sm (142 KiB) downloaded 50 time(s).
Integrate Compendium_21 Simpson-Romberg Merit.sm (138 KiB) downloaded 47 time(s).