Re:hyperbolic substitutions in Calc II, there is a book available on Google books for free called Integration Using Trigonometric and Imaginary Substitutions. It teaches substitutions that we now routinely do using hyperbolic functions using imaginary substitutions, e.g. x = i*sin(theta). It overlooks a lot of complex analysis complications, but is really interesting if you have never come across it. I bet it would be right up your alley.

Looking forward to part 2. If you have any rules of thumb as to which Calc Ii trig substitution integrals are actualy easier using hyperbolic substitution that you teach students, I hope you will share them. I’m teaching this stuff myself this summer.

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]]>Does a similar thing work for functions of two variables? I’m thinking of things that are particularly tough to sketch by hand, like Descarte’s Folium (x^3 + y^3 = 3xy). I have read articles that suggest using something called Newton’s Polygon to sketch these, but I will admit I have not been able to follow the steps that clearly. (My intuition is that this method will not generalize to higher dimensions, because I guess you would have to use partial derivatives, but I am often wrong.)

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