I wrote a new Lecture entirely dedicated to hyperbolic functions and their inverses. In it, I explain what hyperbolic functions are, when – for example – they are useful, and why they sometimes appear in Wolfram results, generally causing panic among students. However, the hyperbolic sine isn’t that scary – I invite you to the Lecture.
Professors at universities have their own requirements. Many of them, for the good of their students, of course, will not hesitate to specify the rules for solving problems in great detail.
Some do not recognize ready-made formulas for some indefinite integrals. See where they came from.
As a curiosity, I’ll calculate the limit of the function: (x^x^x – cosx).
More interesting than the calculations themselves are the two morals that can be drawn from them. But the morals are at the end (has anyone ever seen a story with a moral at the beginning)?
Let’s say we need to calculate the volume of an ellipsoid: {x^2}/4+{y^2}/5+{z^2}/9=1. This is an ellipsoid that intersects the x, y, z axes at coordinates 2, \sqrt{5}, and 3, respectively.
This is not a rotational ellipsoid, it is not formed by rotating any curve around any axis, so we can’t use the standard formula for the volume of a rotational solid. We need to figure out another way.
There are days when nothing seems to work. And there are examples with complex numbers where nothing goes right. The known and memorized methods don’t help.
Take, for instance, this innocent-looking exponentiation: (1+2i)^8. Following the well-trodden path you’ve used in many examples, you want to write the number 1+2i in trigonometric form and then raise it to the eighth power using the appropriate formula. But you encounter complications along the way… See the trick I use.
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