How was the table of basic integrals obtained?

Question

Programming

How was the table of basic integrals obtained? Where did it come from, for example, that the integral of 1/x = ln(x) ?

Answer 1

To begin with, the definition of the derivative is introduced, as the limit of the ratio of the increment of the function to the increment of the argument, with the increment of the argument tending to zero. The main derivatives are derived: hijos.ru/izuchenie-matematiki/mat-analiz-11-klass/...
Next, the antiderivative is introduced as a function inverse to the derivative
Well, and the indefinite integral is defined as the set of all antiderivatives in the form "antiderivative + constant". And in the definite integral, the constant is simply subtracted from itself. So it goes. Read the beginning of mathematical analysis, Ilyin, Sadovnichy, Sendov, for example.

Answer 2

Just like the multiplication table. lower operation.