- #1

As shown in the text, an irreducible quadratic factor

*q = dx*raised to

^{2}+ ex + f*n*in the denominator of a rational function needs to be taken apart as follows:

(a

_{1}x+b

_{1})/q + (a

_{2}x+b

_{2})/q

^{2}+ ... + (a

_{n-1}x+b

_{n-1})/q

^{n-1}+ (a

_{n}x+b

_{n})/q

^{n}

So the problem of integrating all rational functions relegates to solving integrals of the following form:

(ax+b)/q

^{m}(m is natural number)

With a little insight, you can reduce this a little further by adding and subtracing a constant to make it partially amenable to a u-substitution:

(ax+b +c -c)/q

^{m}(c such that ax+b+c is k*q')

(ax+b+c)/q

^{m}- c/q

^{m}(where left integral can be solved with u-sub)

I don't see how to reduce this last form (c/q

^{m}where c constant and q quadratic) despite the claim by the book that it is integrable using methods already known. Does anyone else know (or can figure out) how it's possible to integrate this form in the general case?