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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…).