An alternative to partial fractions

I've been thinking about integration rules, as I've got up to that part of an elementary calculus course for engineering students.  I'm always in two minds about teaching integration rules: do they help the students in any way, or is it just algebraic mumbo-jumbo?  I feel that for students of mathematics, and who wish to understand mathematics to any depth, wrestling with integration rules is helpful not only for algebraic practice, but in getting a deeper understanding of the behaviour of functions.  For students of engineering, however, for whom mathematics is a tool and not an end in itself, I doubt their value.  I'd rather teach the concepts of integration (with pictures), and use a symbolic system for any calculation.  However, at this stage I'm stuck with an old-fashioned course which includes integration rules.  So I'll teach 'em.

This brings me to partial fractions.  In an elementary course, I'd tend not to go much further than fractions whose denominator can be expressed as a product of two linear functions; that is, given a fraction

[

frac{Q}{(ax+b)(cx+d)}

]

where mbox{deg}(Q)le 1, we aim to find A and B for which

[

frac{Q}{(ax+b)(cx+d)}=frac{A}{ax+b}+frac{B}{cx+d}.

]

In a first course, I might mention how to work with more complicated denominators, but in terms of practice I'd stick with easily factorizable quadratics.

If we wanted to go further, we might look at examples of the form

[

frac{Q}{V^k}

]

where V is squarefree (say a quadratic not factorizable over mathbb{R}),  We would try to express this fraction in the form

[

frac{A_k}{V^k}+frac{A_{k-1}}{V^{k-1}}+cdots+frac{A_2}{V^2}+frac{A_1}{V}

]

where each A_i is a linear function of x.  The algebra of such a thing can be monstrous.

However! we can do better than this, using a technique called Hermite reduction which I have learned about in the writings of Manuel Bronstein (1963–2005), who among his many achievements, was the first to implement a complete version of the Risch algorithm for integration of elementary functions.

If V is square-free, then using Euclid's algorithm we can find B and C for which

[

BV'+CV=frac{Q}{1-k}.

]

Then:

[

int frac{Q}{V^k},dx=frac{B}{V^{k-1}}+intfrac{(1-k)C+B'}{V^{k-1}},dx.

]

This last expression is the Hermite reduction formula.

For example, consider the integral

[

intfrac{2x^5+9x^4+16x^3-27x^2-104x-168}{(x^2+2x+5)^3},dx.

]

Nasty! 

We have V=x^2+2x+5 and so V'=2x+2.  Since V is squarefree, Euclid's algorithm will enable us to find polynomials B_0 and C_0 for which

[

B_0V'+C_0V=1.

]

For polynomials as simple as these we could also simply equate coefficients and so find

[

B_0=-frac{x}{8}-frac{1}{8},quad C_0=frac{1}{4}.

]

Then the values B and C can be obtained by

begin{align*}

B&=B_0frac{Q}{1-k}bmod{V},\

C&=frac{Q/(1-k)-BV'}{V}.

end{align*}

In our case

begin{align*}

B&=3x+2\

C&=-x^3-frac{5}{2}x^2+2x+16.

end{align*}

This means that the integral is reduced to

[

frac{3x+2}{(x^2+2x+5)^2}+intfrac{2x^3+5x^2-4x-29}{(x^2+2x+5)^2},dx.

]

Now we apply Hermite reduction to the integral on the right, with the numerator being the new value of Q, and k=2.  The new values of B and C can be found by the formulas above:

begin{align*}

B&=B_0frac{Q}{1-k}bmod{V}=-3x+5\

C&=frac{Q/(1-k)-BV'}{V}=-2x+5

end{align*}

and so the integral is now reduced to

[

frac{3x+2}{(x^2+2x+5)^2}+frac{-3x+5}{x^2+2x+5}+intfrac{2x-2}{x^2+2x+5},dx

]

and this is as far as reduction will take us.  The final integral is easily computed by standard means:

[

intfrac{2x-2}{x^2+2x+5},dx=intfrac{2x+2}{x^2+2x+5}-frac{4}{x^2+2x+5},dx=log(x^2+2x+5)-2arctanleft(frac{x+1}{2}right).

]

Putting all this together, the original integral has been computed to be

[

frac{3x+2}{(x^2+2x+5)^2}+frac{-3x+5}{x^2+2x+5}+log(x^2+2x+5)-2arctanleft(frac{x+1}{2}right)

]

and by a method which, if maybe fiddly to do by hand, is clearly superior to the algebraic horrors which partial fractions would produce.

It is methods like these which form the heart of symbolic integration routines in modern software.  Why then, given that we'd teach a simplified partial fractions technique anyway, do we not ditch partial fractions for something along these lines (but simplified)?

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