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.
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…).
I get a lot of questions about the formula for the canonical form of a quadratic function introduced in my Course on Indefinite Integrals. Why is there a^2? Let me explain.
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