On Geometric Interpretations of Euler’s Substitutions

Round 1

Reviewer 1 Report

Comments as for some modifications as detailed in attached report.

Comments for author File: Comments.pdf

to be modifies as in attached report.

Author Response

Thank you very much for your positive report and several constructive suggestions and corrections. All of them were taken into account.

Reviewer 2 Report

In this under review paper, a classical case of irrational integrals involving the square root of a quadratic polynomial is examined. It is a well-known fact that these integrals can be expressed in terms of elementary functions through Euler's substitutions. Intriguingly, the historical attribution of these three substitutions to Euler remains unclear. In his original work, Leonhard Euler employed only two substitutions, which, remarkably, suffice to cover all possible cases. This analysis sheds light on these intriguing mathematical intricacies, offering a new perspective on the foundations of these classical integrals.

The results are new and correct. However, the paper needs some revision in terms of presentation.     

1) Line 21: It should be "best" instead of "beest".

2) It should be a dot at the end of equation (15). There is the same problem in several equations such as (16), (20), (23), (25),  etc

3) Figures should be presented in smaller sizes.

4) Line 122: It should be "Fig. 8" instead of "Fig 8"

5) The references list should be improved and the authors should mention some recent published papers.

Author Response

Thank you very much for your positive report. All your remarks were taken into account. In particular, new recent references are added and figures are of smaller size (we hope, however, that the size of figures will be suitably adjusted by a technical editor, if needed).