Posted by **mathstudent** on Thursday, January 10, 2008 at 2:41pm.

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?

- Calculus -
**Damon**, Thursday, January 10, 2008 at 3:43pm
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.

- Calculus -
**Count Iblis**, Thursday, January 10, 2008 at 6:36pm
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).

## Answer this Question

## Related Questions

- Pre-Calculus - Find the inverse function for the function f(x)=2^(x)+1/2^(x)-1 ...
- Competetion help on Integration - If there four options are given to a Defenite ...
- algebra - solve the polynomial y=2x^2-4x+1 I am sure neither your text nor your ...
- College Algebra - 1.Give a quadratic function q(x) with a vertex of (-1, 2). 2. ...
- Calculus, check my answers, please! 3 - Did I get these practice questions right...
- Math - Some rational numbers are integers. True or false The radius of a circle...
- Math - True or false Some rational numbers are integers. The radius of a circle ...
- Calculus - Integrate x dx/(1-x). I have proceeded thus- Int xdx/(1-x)=int -(x-1+...
- Calculus - Functions? - #1. A cubic polynomial function f is defined by f(x) = ...
- cal - find the equation of a quadratic function whose graph is tangent at x=1 to...