How Do We Calculate an Integral of a Rational Function from Zero to Infinity?

[zetafunction]

how could we calculate the follwing integral ??

$$\int_{0}^{\infty} \frac{ K(x)}{Q(x)}dx$$

here K(x) and Q(x) are POLYNOMIALS , of course if we had an integral over all R instead of $$(0 , \infty )$$ we could apply Cauchy's residue theorem

i think there is a 'closed circuit' to perform the integral and you have to add a term logx inside the denominator but not completely sure.

 

[Hurkyl]

You could just find the partial fractions decomposition, which is easy to integrate.

(I'm pretty sure you can use residues to help find the decomposition if you like)

 

[HallsofIvy]

If you really want to use residue theory, take the integral from 0 to R, then the circular arc from R to Ri, then down the imaginary axis to 0. If K and Q are reasonably well behaved, you should be able to relate the value on the imaginary axis to its value on the real axis.

If not, break the function into "even" and "odd" parts and integrate each along the real axis from 0 to R, along the circular arc from R to -R and then from -R to 0.

 

[paulfr]

One procedure is this for K(x) / Q(x)...

1/ If the degree of K is = or > than Q, first do Long Division.

2/ With the resulting rational function [remainder / Q] first look
to factor Q and use Partial Fraction Decomposition

3/ If Q is prime and its degree (n) = 2, Complete the Square and use Trig Substitution

4/ If degree of Q is 3 or more, then numerical techniques are needed unless it factors.