# Unsolved Elementary Integrals

I am currently in Integral Calculus, and I wondered if I could get a little creative with my practice. I was curious if there were any unsolved, but rather simple (solvable using methods taught in calc I/II), so that I could be practicing the integration tehniques I have learned while also solving a problem that hasn't been solved yet.

I have consulted lists of integrals, but those are in a more general form (constants are represented by letters, etc.), and I was wondering if there was a place I could find lists of indefinite integrals that had never been calculated but weren't that difficult.

Also, on a side note, if you happen to know any good resources for reading about more techniques of integration, I am currently looking to expand my "toolbox" of integration methods.

Thank you all, and I hope each and every one of you is having a nice day.

• Try (one of my favorites for calc I/II, it uses a trick that not many students are aware off): $$\int \frac{1}{a-\sin(x)}dx, \quad a<1$$ – Rebellos Jan 13 '18 at 0:38
• I'm not sure how one would even go about compiling a list of simple integrals that haven't been solved yet. Say I have an unsolved integral. In order to add it to the list, I would need to also verify that it has a simple solution. If I succeed, the problem stops being unsolved, but if I fail then the problem stops being simple! ;) – David H Jan 13 '18 at 4:14
• @Rebellos For $a=1$ there is a really elementary "cheat" and a very quick solution. But for $a<1$, I can't see a way other than a targent half angle substitution (Weierstrass sub). And I got the answer but it's tedious, involving partial fractions. Is there a better "trick"? – Deepak Jan 14 '18 at 14:28
• @Rebellos For what it's worth I get $\displaystyle \int \frac 1{a-\sin x} dx = \frac 1{\sqrt{1-a^2}}\ln \frac{a\tan\frac x2-1-\sqrt{1-a^2}} { a\tan\frac x2-1+\sqrt{1-a^2}} +c$ – Deepak Jan 14 '18 at 14:51

Here's a relevant flow chart, because as Silvanus P Thompson said:

"Here note this very remarkable fact, that we could not have integrated in the above case [$y = a\log_c x + C$] if we had not happened to know the corresponding differentiation. If no one had found out that differentiating $\log_c x$ gave $x^{-1}$, we [would] have been utterly stuck by the problem [of] how to integrate $x^{-1} dx$. Indeed it should be frankly admitted that this is one of the curious features of the integral calculus :- that you can't integrate anything before the reverse process of differentiating something else has yielded that expression which you want to integrate."

Integration, being a inverse process, is much less systematic than differentiation.

EDIT Right after that quote the author gives an 'unsolved, but rather simple' integral. It may not be solvable though - that quality doesn't necessarily follow from simplicity when it comes to integration.

• I agree with what you said in the edit.. That is an interesting flow chart though, and it does well to provide a more systematic approach to integration. Thank you! – Mike H Jan 13 '18 at 17:37

Some texts that cover the various techniques of integration at a level comparable to Calculus I and II (and a bit beyond) include:

1. The Soviet text Problems in Mathematical Analysis edited by Boris Demidovich has many problems using the various techniques of integration to solve. A few of the techniques Demidovich gives are not usually found elsewhere.

2. Joseph Edwards’ A Treatise on the Integral Calculus (Volume 1) is a particularly valuable source for many interesting integrals, and being published before 1923, is out of copyright meaning it can be readily found online.

3. Michael Spivak’s Calculus contains many interesting questions that use integration, like the proof that $\pi$ is irrational for example.

4. While it is a bit old, G. H. Hardy’s A Course of Pure Mathematics, is also a useful source for questions, and being published before 1923 means it too is no longer under copyright.

5. The recent text How to Integrate It: A Practical Guide to Solving Elementary Integrals by Seán M. Stewart has individual chapters devoted to a particular technique of integration, with each being accompanied by a wealth of end-of-chapter exercises.

• Thank you, I will refer back here plenty I am sure :) – Mike H Jan 13 '18 at 17:38