Delta equals zero in rational indefinite 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.
Factoring a Quadratic Trinomial
In rational indefinite integrals, we often need to factor a quadratic trinomial: 
. We do this of course using the formula: 
, which works when 
.
Rational Integrals and Delta Equals 0
But how does this binomial look when Delta is precisely 0? For instance, how does this factor look: 
?
Is it like this: 
?
Of course not… From high school, we remember that if 
then we get one root, but it’s a double root. So in our example, we can say: 
, meaning the quadratic trinomial factored looks like this:
This has significant consequences for rational indefinite integrals when breaking them into simple fractions.
Example
Let’s take an example:
We break down the fraction by itself without the integral, writing:
We factor out an x in the denominator:
From the quadratic trinomial in the denominator, we compute the delta, which is 0, and get a root of (-1). Factoring it, we get:
And breaking it down into simple fractions:
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