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Canonical Form of a Quadratic Function in Rational Integrals

Krystian Karczyński

Founder and General Manager of eTrapez.

Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.

He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.


I get a lot of questions about the formula for the canonical form of a quadratic function introduced in my Course on Indefinite Integrals.

The formula:

appears in the scheme for solving rational integrals in Lesson 5 of the course:

Scheme for rational integrals

Why is there a^2?

The problem is that at first glance, it looks different from the canonical form known from high school:

The standard question here is: “Why do you have in the denominator?”

Transforming the formula

It’s enough to notice that if in the formula:

we multiply by the square bracket, we get exactly the formula:

(after multiplying by the term it cancels out and we get )

Therefore, both forms are equivalent, which means simply:

So why introduce this formula with the square bracket and a outside the bracket? Because in rational integrals, it is more convenient 🙂

In the later stages of calculating the integral, you will still need to extract in front of the integral sign (and first in front of the bracket in the denominator), so why wait? 🙂


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