I think this can be translated into something not completely wrong. I.e. I have seen calculus like this in old-school books that use operational calculus. It usually uses differential operators instead of integrals, although antiderivatives get operational formulas in terms of the differential operator, and it leads to the Laplace transform because it exhibits identical operational properties.
Why’d the (0) vanish? Everything else seems “justified”.
If anyone’s curious, the first step is wrong. e^x minus its integral is a constant. Not necessarily zero.
Ok. Take the derivative of both sides and integrate at the end, then plug in a value of x to get C = 0.
Yeah I don’t remember if the result is correct, but the process is definitely sus.
the result is correct
Except the first assumption that e^x = its own integral, everything else actually makes sense (except the DX are in the wrong powers). You simply treat the “1” and “integral dx” as operators, formally functions from R^R into R^R and “(0)” as calculating the value of the operator on a constant-valued function 0. So apart from the first error and notational inconsistencies, this is a mathematically correct derivation if you replace 0 with a constant C. EDIT: the step 1/(1-integral) = the limit of a certain series is slightly dubious, but I believe it can be formally proven as well.
