Suppose that ax^2 + bx + c is a quadratic polynomial and that the integration:
Int 1/(ax^2 + bx + c) dx
produces a function with neither a logarithmic or inverse tangent term. What does this tell you about the roots of the polynomial?
well int dx/[x(ax+b)] = 1/b log x/(ax+b)
if c were zero, that is what we would have, and x = 0 would be a root.
So if x = 0 is a root, no good, we get a log.
then int dx/(p^2+x^2) = (1/p)tan^-1 x/p
so we do not want b = 0 either
so we do not want roots of form
x= +/- sqrt (c/a)
That is all I can think of off hand.
First, to simplify things, observe that arctan can be expressed as a logarithm in terms of complex numbers. Suppose we have two different roots (complex or real), y1 and y2. Then:
ax^2 + bx + c = A(x-y1)(x-y2)
for some contant A
We have:
1/[(x-y1)(x-y2)] =
p[1/(x-y1) - 1/(x-y2)]
with
p = 1/(y1-y2)
So, the integral is then clearly a logarithm which can be written as an arctan if the roots are complex. Now any second degree polynomial has two roots in the set of complex numbers, however, the two roots can coincide. If that happens then the polynomial is proportional to:
1/(x-y1)^2
If we integrate this we obtain a term proportional to 1/(x-y1), which is not a logarithm nor an arctan. So, the only way to avoid a logarithmic or arctan term is if the two roots coincide to form a single root (we say that the root has a multiplicity of 2, when counting roots it counts double).
To determine the information about the roots of the quadratic polynomial, we need to analyze the behavior of its integral.
Let's start by considering the integral:
Int 1/(ax^2 + bx + c) dx
When we integrate a rational function like this, the nature of the resulting integral depends on the discriminant of the quadratic polynomial in the denominator, which is given by Δ = b^2 - 4ac. The discriminant determines the types of roots the quadratic polynomial may have.
Now, in this case, we are told that the integral of the function does not have a logarithmic or inverse tangent term. This information means that the discriminant Δ cannot be equal to zero, as it would result in logarithmic or inverse tangent terms in the integral.
Therefore, if the integral has neither a logarithmic nor an inverse tangent term, we can conclude that the discriminant Δ is not zero. In other words, the quadratic polynomial does not have two identical real roots.
In terms of roots, this tells us that the quadratic polynomial either has two distinct real roots or it has a complex conjugate pair of roots.