Int(f(x), x) in MATLAB does not explicitly include the +C (an arbitrary constant that must be included when integrating).
The output of the heavyside function is 0 if negative, 1 if positive. The output of the sign function is 1 if positive and -1 if negative. if you divide the sign function output range by two your possible results become [-1/2, 1/2].
Adding 1/2 to this set of possible answers produces [0, 1], identical outputs to the heavyside function. The 1/2 that we added is included in that implicit +C.
Please run the below command in the command window of installed MATLAB R2019a version to get release specific documentation on the implicit +C for the "dirac" function:
>> web(fullfile(docroot, 'symbolic/dirac.html'))
As we can see, in the context of integrals, the heavyside function and the sign function divided by 2 are equivalent when each has an arbitrary constant. These are both equally valid solutions.
Please follow the below link to search for the required information regarding the current release:
