I’ve added a new article with a video to the “Lectures” section (on the right sidebar): Cyclometric Functions.
It’s dedicated to cyclometric functions (arcsinx, arccosx, arctgx, arcctgx) and is divided into two parts. In the first part, I quickly show how to calculate them, and in the second part, I delve deeper into the topic (definitions, graphs).
In previous posts, I demonstrated how to use Euler substitutions in integrals with the square root of the polynomial ax^2+bx+c.
Euler substitutions of the first kind were used when a>0, and second kind Euler substitutions when c>0. In this post, we will deal with the third and final kind of Euler substitutions that can be used when the quadratic polynomial in the integral has TWO DISTINCT roots x1, x2, meaning its discriminant is positive. See what to do in this case.
In the previous post: Euler Substitution of the First Kind, we dealt with integrals involving the root of the trinomial {ax^2+bx+c}, where a>0.
But what if “a” in the trinomial is negative? Then the second kind of Euler substitution, for c>0, might help us (but not necessarily…).
When calculating square roots in Cartesian (or: algebraic) form in my Complex Numbers Course, I showed a method that involves adding a third equation to the existing system of two equations, which greatly simplifies and shortens the further calculations.
I showed this method, but didn’t justify it in any way. Recently, I received an email regarding this:
“Could you explain why we can use the method of adding a third equation when calculating the square root of a complex number?” So here’s the explanation.
This post is a kind of response to the question:
“I don’t understand something and I need an explanation, why do you simplify ‘n’? I mean that n/n is an indeterminate symbol (infinity over infinity) help because I’m already lost with this.”
Understanding what indeterminate symbols REALLY are can be quite tricky. It also raises many questions about what you “can” and “can’t” do with them.
