# The hardest case of integration by partial fractions

The context is explaining to calculus students how to integrate rational functions by using partial fractions decomposition.

As we all know, partial fractions decomposition is a method to write every rational function as a sum of rational functions of the following three types: $$(1)\text{ polynomial}, \quad(2)\;\frac{A}{(ax + b)^n}, \quad(3)\;\frac{Ax + B}{(ax^2 + bx + c)^n},$$ where $$n$$ is a positive integer and $$A,B,a,b,c$$ are real numbers such that $$a \neq 0$$ and $$b^2 - 4ac < 0$$.

After having done it many times, I can say that I have no trouble explaining how to do partial fractions decomposition and that it is usually understood quite well by the students. Also, I have no trouble explaining how to integrate (1), (2), and for $$n = 1$$ (3), which are the cases that usually appear in exams.

However, I always find difficult explaining how to integrate (3) for a general $$n \geq 1$$. This of course can be done by induction on $$n$$, using the third-last and second-last formulas here, but those formula are essentially impossible to memorize.

Is there a better method, that requires little memorization for the students?

• Other than maybe a long'ish homework project done for mental toughness training, fully carrying out the process in which one of the terms is your (3) for $n \geq 2,$ I wouldn't worry about (3). In practice, at least before computer algebra systems were around, you used a reduction formula in an integral table for this (indeed, most calculus texts would have what I'm thinking of in their short back-of-the-book table of integrals). I often discussed the use of such a formula in U.S. Calculus 2 classes, and on occasion I had such a problem (with formula included) (continued) Nov 1, 2019 at 14:42
• on an in-class test, but typically on my in-class tests I would separate out the various skills (otherwise, partial credit for correct work done after making a reasonable error near the beginning makes grading a nightmare): 3 or 4 problems where one shows the correct "set-up form" for partial fraction decomposition and maybe 2 problems in which a "set-up form" is provided (and try to avoid these from cluing how to do the previous problems) to find the various partial fraction coefficients. As for the final indefinite integrations, that was covered by various indefinite integrations problems. Nov 1, 2019 at 14:54
• It seems absurd to me that students are still being force-fed this kind of thing. For the top 1% of students, the thing to do is to teach them partial fractions over the complex numbers, where the theory and results are simple and beautiful, rather than over the reals, where everything is ugly. For the other 99%, it's ridiculous to imagine that they will benefit from this. If they run into such an integral, they'll do it using software. Here in California, the absurdity is further heightened by the fact that UC biology majors now have to take this. Pharmacists do not use partial fractions.
– user507
Nov 1, 2019 at 15:47
• @BenCrowell "For the other 99%, it's ridiculous to imagine that they will benefit from this. If they run into such an integral, they'll do it using software." I agree, but I'm not the one who decides the topics of the exam. Nov 1, 2019 at 16:31
• Is there a better method, that requires little memorization for the students? I would say conceptually the most elegant and also simplest way would be by introducing and working in the complex numbers, where everything splits into linear factors so we don't have to worry about terms of the type (3). But I'm not making this an answer since obviously, it might not be very practical to introduce complex numbers and then the Fundamental Theorem of Algebra purely for the purpose of integration. Nov 17, 2019 at 10:59

Left to my devices, for case (3.) I'd use a trig. substitution or a u-substitution. First, put the expression into the following form: $$\int\frac{ A(x-\alpha)+B}{[(x-\alpha)^2+\beta^2]^n} \, dx$$ The first part isn't bad, nearly all books cover this part, $$\int \frac{ A(x-\alpha)\, dx}{[(x-\alpha)^2+\beta^2]^n} = \int\frac{A \, du}{2u^n}$$ using a $$u=(x-\alpha)^2+\beta^2$$ substitution. So, that part isn't hard. On the other hand, setting $$x-\alpha = \beta \tan \theta$$ gives $$(x-\alpha)^2+\beta^2 = \beta^2(\tan^2 \theta+1)=\beta^2 \sec^2 \theta$$ and $$dx = \beta \sec^2 \theta d \theta$$ thus $$\int \frac{ Bdx}{[(x-\alpha)^2+\beta^2]^n} = \int \frac{ B\beta \sec^2 \theta \, d \theta}{[\beta^2 \sec^2 \theta]^n} = B\beta^{2n-1} \int \cos^{2n-2} \theta \, d\theta.$$ In the case $$n=1$$ we just face $$\int d\theta = \theta$$ and so we recover the usual arctangent solution that is often presented as something to memorize. If $$n=2,3,...$$ then we can just face the integral of cosine using whichever method you prefer. Personally, I like direct battle via repeated substitution of $$\cos^2 \theta = \frac{1}{2}(1+ \cos 2 \theta)$$, but I understand others are partial to the iterative formulas for integration of powers of cosine. In summary, the integral is really not that bad, it's just tedious if the polynomial in the denominator is quartic or higher order.

All of this said, it's probably a bit much for a test unless it was belabored in the homework.

• I love the phrasing "direct battle" as a contrast to the method of using an integration table. Nov 18, 2019 at 15:21

I have 3 calculus books and all of them cover this. Thomas Finney and Swokowski show the method that James Cook showed (complete the square and trig substitution). Granville shows the reduction formula that Renfro referred to, saying that students should practice it a bit to be familiar with it and able to use it out of integral tables, but not required to memorize it. (I prefer the Thomas/Finney explanation the best.)

Thomas Finney shows it well, but has only a single drill problem (only has 20 for all of partial fractions lesson). Swokowski has a bit more drill, couple problems. Granville has eight drill problems for this technique. However, Thomas Finney and Swokowski also have "end of chapter" drill problem sections of over 100 problems, in addition to the "end of lesson" problems. And a few more drill problems on this topic, mixed in there.

I don't mean to sound harsh, but I just read the textbook and worked some homework. It's not any different for a teacher versus a student in building skills. I promise that if you work 10 problems, you'll know it cold. This is a slightly esoteric technique, but that's it. Also, it does give some practice in using other algebraic manipulations. So it's got some integrative (pun intended) benefit. And building manipulative muscles is not a bad thing for stronger STEM students.

There is nothing wrong with (a) not teaching this; (b) just teaching it and not testing it, maybe giving a single homework problem; or (c) teaching it and drilling it hard, until mastery. It all depends on how much time you have to cover how much material, how strong your kids are, and what you want to emphasize. But first learn it yourself.