241

I begin this post with a plea: please don't be too harsh with this post for being off topic or vague. It's a question about something I find myself doing as a mathematician, and wonder whether others do it as well. It is a soft question about recreational mathematics - in reality, I'm shooting for more of a conversation.

I know that a lot of users on this site (e.g. Cleo, Jack D'Aurizio, and so on) are really good at figuring out crafty ways of solving recreational definite integrals, like $$\int_{\pi/2}^{\pi} \frac{x\sin(x)}{5-4\cos(x)} \, dx$$ or $$\int_0^\infty \bigg(\frac{x-1}{\ln^2(x)}-\frac{1}{\ln(x)}\bigg)\frac{dx}{x^2+1}$$ When questions like this pop up on MSE, the OP provides an integral to evaluate, and the answerers can evaluate it using awesome tricks including (but certainly not limited to):

  • Clever substitution
  • Exploitation of symmetry in the integrand
  • Integration by parts
  • Expanding the integrand as a series
  • Differentiating a well-know integral-defined function, like the Gamma or Beta functions
  • Taking Laplace and Inverse Laplace transforms

But when I play around with integrals on my own, I don't always have a particular problem to work on. Instead, I start with a known integral, like $$\int_0^\pi \cos(mx)\cos(nx) \, dx=\frac{\pi}{2}\delta_{mn},\space\space \forall m,n\in \mathbb Z^+$$ and "milk" it, for lack of a better word, to see how many other obscure, rare, or aesthetically pleasing integrals I can derive from it using some of the above techniques. For example, using the above integral, one might divide both sides by $m$, getting $$\int_0^\pi \frac{\cos(mx)}{m}\cos(nx) \, dx=\frac{\pi}{2m}\delta_{mn},\space\space \forall m,n,k\in \mathbb Z^+$$ Then, summing both sides from $m=1$ to $\infty$, and exploiting a well-known Fourier Series, obtain $$\int_0^\pi \cos(nx)\ln(2-2\cos(x)) \, dx=-\frac{\pi}{n},\space\space \forall n\in \mathbb Z^+$$ or, after a bit of algebra, the aesthetically pleasing result $$\int_0^{\pi/2} \cos(2nx)\ln(\sin(x)) \, dx=-\frac{\pi}{4n},\space\space \forall n\in \mathbb Z^+$$ After pulling a trick like this, I look through all of my notebooks and integral tables for other known integrals on which I can get away with the same trick, just to see what integrals I can "milk" out of them in the same way. This is just an example - even using the same starting integral, countless others can be obtained by using other Fourier Series, Power Series, integral identities, etc. For example, some integrals derived from the very same starting integral include $$\int_0^\pi \frac{\cos(nx)}{q-\cos(x)} \, dx=\frac{\pi(q-\sqrt{q^2-1})^{n+1}}{1-q^2+q\sqrt{q^2-1}}$$ $$\int_0^\pi \frac{dx}{(1+a^2-2a\cos(x))(1+b^2-2b\cos(mx))}=\frac{\pi(1+a^m b)}{(1-a^2)(1-b^2)(1-a^m b)}$$ and the astounding identity $$\int_0^{\pi/2}\ln{\lvert\sin(mx)\rvert}\cdot \ln{\lvert\sin(nx)\rvert}\, dx=\frac{\pi^3}{24}\frac{\gcd^2(m,n)}{mn}+\frac{\pi\ln^2(2)}{2}$$ Everyone seems to be curious about the proof of this last identity. A proof can be found in my answer here.

I just pick a starting integral, and using every technique I know as many times as possible, try to come up with the most exotic integrals as I can, rather than picking a specific integral and trying to solve it.

Of course, integrals generated this way would be poor (or at least extremely difficult) candidates for contest problems or puzzles to evaluate given the integral, since they are derived "backwards," and determining the derivation given the integral is likely much harder than pursuing the vague goal of a "nice-looking integral" with no objective objective (ha ha).

QUESTION: Do you (residents of MSE who regularly answer/pose recreational definite integral questions) do this same activity, in which you try to generate, rather than solve, cool integrals? If so, what are some integrals you have come up with in this way? What strategies do you use? Does anyone care to opine on the value (or perhaps lack of value) of seeking integrals in this way?

Cheers!

Franklin Pezzuti Dyer
  • 37,332
  • 9
  • 60
  • 145
  • I once found such an example on AoPs, I will try to find it. It was a carefully constructed integral whose general form does not have a solution. Is this the kind of stuff you are looking for? –  Jun 15 '18 at 22:44
  • @JefferyOpoku-Mensah Yeah, that sounds like what I'm describing. In general, I'm trying to find out what the popular opinion is of "integral construction" as a skill, competed to "integral evaluation." – Franklin Pezzuti Dyer Jun 15 '18 at 22:47
  • 1
    I am curious to know what other identities (not limited to integrals) have been "milked/reverse-derived". – Dair Jun 16 '18 at 06:44
  • 28
    I'd like to object to your claim that these are poor candidates for contest problems. Back in the days, our "trainer" for the IMO sometimes lectured us about ways to obtain nice problems (not necessarily integration related) from some well-known truth by playing around with it long enough to erase most traces to the starting point. The challenge is of course to end up with something that does not look arbitrary and convoluted... Then again, it might have been that the lecture title ways "*Poor* man's problem generation" – Hagen von Eitzen Jun 16 '18 at 09:31
  • 2
    I used to have similar fun deriving trigonometric identities. –  Jun 16 '18 at 11:47
  • Can anyone provide me an example of using Contour Integration to generate results i'm intrigued to try to "Integral Milking now – Zophikel Jun 16 '18 at 14:19
  • 11
    @Zophikel Well, one cool integral derived using the Residue Theorem is this integral involving the Lambert-W function: $$\int_{-\infty}^{\infty} \frac{dx}{(ae^x-x)^2+\pi^2}=\frac{1}{1+W(a)}$$ and what I consider to be my favorite integral of all time involving my very favorite constant, the Dottie Number (denoted $ա$): $$\int_{0}^\infty \frac{3\pi^2+4(z-\sinh z)^2}{(3\pi^2+4(z-\sinh z)^2)^2+16\pi^2(z-\sinh z)^2}dz=\frac{1}{8+8\sqrt{1-ա^2}}$$ which I derived at the following link: https://math.stackexchange.com/questions/2446725/integral-representation-of-the-dottie-number – Franklin Pezzuti Dyer Jun 16 '18 at 14:25
  • @Frpzzd so in general if you want to milk an integral for a function that has poles of course over a given Contour one would have to use the Residue Theorem, are there any examples that don't exploit the Residue Theorem ? – Zophikel Jun 16 '18 at 14:28
  • @Zophikel Most integrals (that I know of) that are derived using Countour Integration use the Residue Theorem, or at least the same tricks that the Residue Theorem uses (like cutting keyholes out of contours) paired with some other tricks (like letting the contour grow infinitely large). I would check out the book "Inside Irresistible Integrals," since the author demonstrates a couple of derivations of integrals that use Contour Integration without the Residue Theorem per se. – Franklin Pezzuti Dyer Jun 16 '18 at 14:46
  • 1
    @Dair Well, the reason I asked this question is that I realized that most of my blog posts about definite integrals, though they pretended to be about methods of integral *evaluation*, were *actually* about methods of integral milking. Take a look if you want: http://franklin.dyer.me – Franklin Pezzuti Dyer Jun 16 '18 at 15:19
  • Cleo was suspicious af. idk if anyone ever figured up what was her deal. – qwr Jun 16 '18 at 19:27
  • @qwr Suspicious? I'd use the word *mysterious.* And who knows how she does what she does... I would be delighted (and surprised) if she commented on this question. – Franklin Pezzuti Dyer Jun 16 '18 at 19:29
  • 5
    "I'm shooting for more of a conversation." This site is not for conversation. Voting to close. – bof Jun 16 '18 at 19:35
  • 5
    @bof Oh, I'm sorry to hear that. I've just been itching to ask other mathematicians about this phenomenon, but I don't know any other way to access such a diverse community of math-lovers than through MSE. – Franklin Pezzuti Dyer Jun 16 '18 at 21:15
  • 4
    A wonderful perspective. I have done this in the past but never made it my recreation as it seems you have. The gcd integral is quite amazing indeed, especially when you plot the integrand over the domain of integration for different values of $m$ and $n$. Truly astounding ! – gone Jun 17 '18 at 08:17
  • 17
    You might want to publish on your "astounding identity". I've seen papers looking for novel algorithms to compute GCD in parallel that rely on integration techniques. Just a thought. – COTO Jun 17 '18 at 15:55
  • @bof this question will more likely be locked considering the number of upvotes. – tox123 Jun 17 '18 at 17:17
  • This (or rather the automated equivalent) is useful for finding bugs in CAS systems. Take an equation and permute it in a way that should preserve the identity. If it can't solve it? Meh, make a note and try something else. If it solves it? Great. If it says that e.g. F(x) == G(x) but F(x)/|C+1| != G(x)/|C+1|? Whoops, there's a bug. Also useful for "building blocks" (bigint libraries, etc). – TLW Jun 17 '18 at 17:55
  • Could you please re-consider your use of notation? I find it rather bad in multiple ways: **1.** $\ln^2(x)$ or $\operatorname{gcd}^2(m,n)$ should be avoided. Hasn't the $\cos^2(\varphi)$ vs. $\cos^{-1}(y)$ debacle taught us anything? **2.** The integration $\mathrm{d}x$ (many people keep the $d$ italic, but making it `\mathrm{d}` certainly can't hurt to make it stand out) should be either in right front, or last, or if you like the only thing on top of a fraction integrant, but not _in some fraction somewhere in the integrand_. This becomes a hide-and-seek. – leftaroundabout Jun 18 '18 at 11:43
  • **3.** something like $\ln|\sin(mx)|\ln|\sin(mx)|dx$ is unnecessarily hard to parse; why not throw in some `\cdot`s, spacing, and/or parentheses to disambiguate between multiplications and function applications? – leftaroundabout Jun 18 '18 at 11:43
  • @leftaroundabout Sorry for the bad formatting; I will see what I can do! :) I'm afraid I differ with you, though, on the use of $\ln^2$ and $\gcd^2$... the former is pretty commonly used and understood. – Franklin Pezzuti Dyer Jun 18 '18 at 15:40
  • I’ve long suspected that if you look in a big enough book of integrals, at least some of the results will have been “milked” from others. – David K Jun 18 '18 at 16:52
  • 3
    I would _love_ to see a proof of that (truly) astounding identity. – fmt Jun 18 '18 at 17:24
  • 5
    I am surprised nobody mentioned Integrators Anonymous. The have an $n$-step program, where $n \to \infty$. – copper.hat Jun 19 '18 at 20:54
  • I would also like to see a proof of this identity. I've posted this as a separate question [here](https://math.stackexchange.com/questions/2826571/an-astounding-identity-int-0-pi-2-ln-lvert-sinmx-rvert-cdot-ln-lvert). – Jair Taylor Jun 20 '18 at 21:05
  • 12
    [From our Help Center](https://math.stackexchange.com/help/dont-ask) *If your motivation for asking the question is “I would like to participate in a discussion about ______”, then you should not be asking here.* Therefore you paint yourself into a corner with your first paragraph. This would make for an interesting conversation, but it makes a poor question in MSE.* – Jyrki Lahtonen Jun 22 '18 at 10:09
  • 6
    I'm voting to close this question as off-topic because I agree with Jyrki Lahtonen's assessment---this is an interesting discussion which could be had, but is a bad question for MSE. – Xander Henderson Jun 27 '18 at 01:31
  • @amwhy: I'm afraid I share said lack of life... – copper.hat Jun 27 '18 at 22:19
  • 1
    @copper.hat I appreciate and feel honored that you felt safe enough to disclose that info to me, but perhaps you should consult professional help? – amWhy Jun 27 '18 at 23:39
  • @amWhy: Thank you for your concern, it is much appreciated; my comment was perhaps more prosaic than you interpreted - I was bemoaning usual first world problems such as teens, commuting, ever shrinking currency and lack of available time. MSE has become a source of entertainment that can be slotted into the otherwise unproductive gaps. It would be fair to comment that many friends have suggested professional help, but I think usually in the spirit of frustration :-). – copper.hat Jun 28 '18 at 02:56

11 Answers11

152

Yes, definitely. For example, I found that $$ m\int_0^{\infty} y^{\alpha} e^{-y}(1-e^{-y})^{m-1} \, dy = \Gamma(\alpha+1) \sum_{k \geq 1} (-1)^{k-1} \binom{m}{k} \frac{1}{k^{\alpha}} $$ (and related results for particular values of $\alpha$) while mucking about with some integrals. Months later, I was reading a paper about a particular regularisation scheme (loop regularisation) useful in particle physics, and was rather surprised to recognise the sum on the right! I was then able to use the integral to prove that such sums have a particular asymptotic that was required for the theory to actually work as intended, which the original author had verified numerically but not proved. The resulting paper's on arXiv here.

Never let it be said that mucking about with integrals is a pointless pursuit!

Chappers
  • 65,147
  • 11
  • 62
  • 122
  • 21
    Wowie, that is certainly a cool one... and what a coincidence to run into it later! This certainly demonstrates (coincidental) value in "mucking about" with integrals! – Franklin Pezzuti Dyer Jun 15 '18 at 23:03
  • 4
    Rearranging your formula slightly and summing once again after incorporating factor $\frac{1}{2^m}$ appears to give $$\sum_{m=1}^\infty\frac{1}{2^m}\frac{1}{\Gamma(\alpha+1)}\int_0^{\infty} y^{\alpha} e^{-y}(1-e^{-y})^{m-1} \, dy = \sum_{m=1}^\infty\frac{1}{2^m}\frac{1}{m} \sum_{k \geq 1} (-1)^{k-1} \binom{m}{k} \frac{1}{k^{\alpha}}=\eta(1+\alpha)$$ where $\eta(0)=\log2$, $\eta(1)=\pi^2/12$ and more generally $\;\eta(1+\alpha)=(1-2^{1-n})\zeta(1+\alpha)$ – James Arathoon Jun 17 '18 at 12:23
  • 3
    sorry in above comment $\eta(0)$ should read $\eta(1)$ and $\eta(1)$ should read $\eta(2)$ and $2^{1-n}$ should read $2^{1-\alpha}$ – James Arathoon Jun 17 '18 at 12:31
  • 1
    @JamesArathoon That looks like the [Euler transform](http://mathworld.wolfram.com/EulersSeriesTransformation.html) of the $\eta$ series. – Chappers Jun 17 '18 at 12:43
49

Unsure if this is worthy of an answer, but one particular trick I find fascinating is coordinate changes that leave the result of an integration untouched.

For example, there's a theorem with a name I can't remember right now (EDIT: It's called Glasser's master theorem, as Chappers pointed out below) that establishes equivalence of integrals of real functions over the entire real line:

$$\int_{-\infty}^{\infty}f(x)dx = \int_{-\infty}^{\infty}f\left(|\alpha|x - \sum_{i=1}^{n}\frac{|\gamma_i|}{x - \beta_{i}}\right)dx$$

for arbitrary constants $\alpha$, $\beta_i$, $\gamma_i$.

The reason why this is great for "milking" integrals is that you can keep changing the coordinates over and over until you get a monstrosity that has a simple result.

Let's try the simplest example I can think of, $\int_{-\infty}^{\infty}\frac{1}{x^2 + a}dx$ with real positive $a$. Then, by applying the coordinate change over and over using $\alpha = 1$, $\gamma_{i} = \gamma =1$ and $\beta_i = \beta = 0$:

$$\frac{\pi}{\sqrt{a}}=\int_{-\infty}^{\infty}\frac{1}{x^2 + a}dx = \int_{-\infty}^{\infty}\frac{x^2}{x^4 + (a+2)x^2 + 1}dx = \int_{-\infty}^{\infty}\frac{(x^2 (x^2 + 1)^2)}{a x^6 + 2 a x^4 + a x^2 + x^8 + 6 x^6 + 11 x^4 + 6 x^2 + 1} dx= \quad...$$

I'm certainly not suggesting this is a difficult integral, but you can see how it can get very hairy if I had nonzero $\beta_i$'s or more than one $\gamma_i$!

Once you have enough of these types of transforms under your belt, you can apply them to your heart's content in whatever form you like knowing that the result remains unchanged.

Hope this helps your quest to find more wonky integrals!

John Bentin
  • 16,449
  • 3
  • 39
  • 64
aghostinthefigures
  • 2,852
  • 1
  • 10
  • 23
  • 13
    It's often called [Glasser's Master Theorem](http://mathworld.wolfram.com/GlassersMasterTheorem.html), which is a pretty terrible name, since it seems to have been known in various forms for a couple of hundred years. – Chappers Jun 16 '18 at 00:06
  • 8
    @aghostinthefigures Haha, yeah, that's one of my favorite integral theorems. One result that I've "milked" using Glasser's Master Theorem was $$\int_0^\pi \frac{\sin^2(2^n x)}{\sin^2(x)}dx=2^n\pi,\space\space \forall n\in\mathbb Z^+$$ using repeated application of the theorem and trig substitution. – Franklin Pezzuti Dyer Jun 16 '18 at 14:20
  • 5
    @aghostinthefigures If you're curious, another thing derived from Glasser's master theorem is the following: if $n$ is an integer, then $$\int_{-\infty}^\infty f(x)dx=\frac{a}{2^n}\int_{-\pi/2}^{\pi/2} f(a\cot(2^n x))\frac{dx}{\sin^2(x)}$$ of which the integral in the comment above is a special case, with $a=1$ and $f(x)=\frac{1}{1+x^2}$. – Franklin Pezzuti Dyer Jun 16 '18 at 22:32
  • 3
    Fascinating; I'm glad my input helped you find some great integrals! It's worth mentioning that I usually employ these sorts of "backwards" coordinate change tricks on ODE/PDE's, to see if I can dig up a series of transforms that make some nasty nonlinear equation linear. But those are a story for another post, I suppose. – aghostinthefigures Jun 17 '18 at 04:25
38

Mathematicians milk all sorts of things just for the fun of it!

  1. One Jon concocted a sequence of integrals using Fourier analysis that at first seem to evaluate to $\pi/2$ and then break down at the seventh term. He reported it as a bug to a computer algebra system vendor, who spent 3 days before figuring out he had been had.

  2. That whole MO thread above is about coming up with counter-examples to computer algebra systems such as difficult-to-see convergence. I also do that do Wolfram Alpha all the time; it will forever remain possible to easily trick it to give the Wrong Answer.

  3. The 1968 Putnam competition featured an obviously positive integral that evaluates to $\frac{22}{7}-\pi$. Lucas concocted another similar integral for $\frac{355}{113}-\pi$.

  4. You could ask the creator of $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi$ how he/she came up with this integral. It does not look like a naturally occurring integral to me.

user21820
  • 53,491
  • 7
  • 84
  • 231
  • First sentence reply: Yes, and they milk all sorts of things, Including reputation on site's like MSE! – amWhy Jun 28 '18 at 15:33
  • @amWhy: I did not have points in mind when posting the answer. But it seems people are generally happy to see these examples. =) – user21820 Jun 28 '18 at 15:56
  • It seems people hare generally happy to see fireworks. But fireworks aren't on topic on MSE, now are they? – amWhy Jun 28 '18 at 16:01
  • 2
    @amWhy: I'm just stating the apparent reason for the upvotes here. As for whether the question itself is off-topic, I have explained my reasons [here](https://chat.stackexchange.com/transcript/message/45372174#45372174) why in my opinion it is. But I will leave it to others to decide that. The history shows that it has gone through 2 cycles of closing and reopening already. – user21820 Jun 28 '18 at 16:10
  • That "one Jon" would be [Jonathan Borwein](https://en.wikipedia.org/wiki/Borwein_integral), wouldn't he? –  Jan 29 '19 at 18:43
  • @Rahul: Yes! See Jacques' comment. =) – user21820 Jan 30 '19 at 11:32
22

We've seen Glasser's Master Theorem from ghostinthefigures well Ramanujan had one too:

Ramanujan's Master Theorem: Ramanujan was especially fond of this technique of his. G. H. Hardy stated that he "was particularly fond of them, and used them as one of his commonest tools (to milk his integrals)." Note that his procedure is strictly formal with results being valid only under severe conditions.

Let $F(x)$ be some complex-valued function in some neighbourhood of $x=0$ with expansion $$F(x)=\sum_{k=0}^{\infty}\frac{\lambda(k)(-x)^k}{k!}$$ where $\lambda$ is some analytic singular valued function. Then $$ \int_{0}^{\infty}x^{n-1}F(x)\,dx=\Gamma(n)\lambda(-n)\tag{1} $$ Alongside his Master Theorem $(1)$ Ramanujan also gave the equivalent identity: $$\int _{0}^{\infty }x^{s-1}(\varphi(0)-x\varphi(1)+x^{2}\varphi(2)-\cdots)\,dx=\frac {\pi }{\sin(\pi s)}\varphi(-s)\tag{2}$$ which gets converted to the above form $(1)$ after substituting $\varphi(n)=\frac {\lambda(n)}{\Gamma (1+n)}$ and using the functional equation for the gamma function.

Ramanujan's proof starts with Euler's gamma function integral $$\int_{0}^{\infty}e^{-mx}x^{n-1}\,dx=m^{-n}\Gamma(n),\quad m,\,n>0$$ Let $m=r^k$, $r$ a constant $>0$, and multiply both sides by $f^{(k)}(a)h^k/k!$, where $f$ is some function to be defined later with $a$, $h$, constants. Now sum on $k$ thus: $$ \sum_{k=0}^{\infty}\frac{f^{(k)}(a)h^k}{k!} \int_{0}^{\infty}e^{-r^{k} x}x^{n-1}\,dx =\Gamma(n)\sum_{k=0}^{\infty}\frac{f^{(k)}(a)(hr^{-n})^k}{k!} $$ Now he expands $e^{-r^{k} x}$ in its Maclaurin series, inverts the order of summation and integration to get \begin{align*} \sum_{k=0}^{\infty}\frac{f^{(k)}(a)h^k}{k!} & \int_{0}^{\infty} x^{n-1}\sum_{j=0}^{\infty}\frac{r^{jk}(-x)^j}{j!}\,dx = \int_{0}^{\infty} x^{n-1}\sum_{j=0}^{\infty}\frac{(-x)^j}{j!}\sum_{k=0}^{\infty}\frac{f^{(k)}(a)h^k r^{jk}}{k!}\,dx\\ &=\int_{0}^{\infty}x^{n-1}\sum_{j=0}^{\infty}\frac{f(a+hr^{j})(-x)^j}{j!}\,dx=\Gamma(n)f(a+hr^{-n})\tag{3} \end{align*} For $m$ real, let $f(hr^{m}+a)=\lambda(m)$, then $(3)$ can be written as $(1)$.

As an example of $(1)$ consider the binomial expansion $$(1+x)^{-a}=\sum_{k=0}^{\infty}\binom{k+a-1}{k}(-x)^k =\sum_{k=0}^{\infty}\frac{\Gamma(k+a)}{\Gamma(a)}\frac{(-x)^k}{k!}$$ Let $\lambda(k)=\Gamma(k+a)/\Gamma(a)$. Then Ramanujan’s Master Theorem gives $$\int_{0}^{\infty} \frac{x^{n-1}}{(1+x)^a}\,dx=\frac{\Gamma(n)\Gamma(a-n)}{\Gamma(a)}=B(n,a-n)$$ where $B$ is the beta integral.

As an example of $(2)$ consider the infinite product definition of the Gamma function: $$\Gamma (x)=\frac {e^{-\gamma x}}{x}\prod _{n=1}^{\infty}\left(1+\frac {x}{n}\right)^{-1}e^{x/n}$$ is equivalent to the expansion $$\log \Gamma (1+x)=-\gamma x+\sum _{k=2}^{\infty }\frac {\zeta (k)}{k}(-x)^{k}$$ where $\zeta (k)$ is the Riemann zeta function. This rearranges to $$\frac{\gamma x+\log \Gamma (1+x)}{x^2}=\sum _{k=0}^{\infty }\frac {\zeta (2+k)}{2+k}(-x)^{k}$$ Let $\varphi(k)=\zeta (2+k)/(2+k)$. Then applying Ramanujan’s Master Theorem we have: $$\int _{0}^{\infty }x^{s-1}\frac {\gamma x+\log \Gamma (1+x)}{x^{2}}\,dx=\frac {\pi }{\sin(\pi s)}\frac {\zeta (2-s)}{2-s}$$ valid for $0<\Re(s)<1$.

(See Ramanujan's Notebooks Vol I, Ch.4 for some of Ramanujan's results, and Hardy's Ramanujan Ch.XI, Definite Integrals, for thorough proofs.)

Daniel Buck
  • 3,326
  • 1
  • 10
  • 27
15

I'm not sure if this is what you are looking for, but I found the following (elliptic?) integral on AoPS:

$$\int \frac{x}{\sqrt{x^4+4x^3-6x^2+4x+1}}\, \mathrm{d}x$$

It was likely found by studying the Risch algorithm, or at least the Wikipedia page. Here it can be seen the general form of this integral cannot be solved in elementary terms, but by carefully choosing the right polynomials, one can construct a function whose derivative is of that form.

As shown here, the solution to the integral is the monstrous

$$ -\frac{1}{6}\ln\left[(x^{4}+10x^{3}+30x^{2}+22x-11)\sqrt{x^{4}+4x^{3}-6x^{2}+4x+1}- \\(x^{6}+12x^{5}+45x^{4}+44x^{3}-33x^{2}+43)\right]+C$$

This stems from the integral given in the Wikipedia page,

$$\int \frac{x}{\sqrt{x^4 + 10 x^2 - 96 x - 71}} \mathrm{d}x$$

which evaluates to

$$ - \frac{1}{8}\ln \,\Big( (x^6+15 x^4-80 x^3+27 x^2-528 x+781) \sqrt{ x^4+10 x^2-96 x-71} \Big. \\ {} - \Big .(x^8 + 20 x^6 - 128 x^5 + 54 x^4 - 1408 x^3 + 3124 x^2 + 10001) \Big) + C$$

  • 4
    Technically these are only [pseudo-elliptic](https://www.encyclopediaofmath.org/index.php/Pseudo-elliptic_integral). – Chappers Jun 15 '18 at 23:05
13

I have not seen anyone else mention Irresistible Integrals by Boros & Moll, so look up that book.

  • 3
    Great book. If you like that, check out Paul Nahin's *[Inside Interesting Integrals](https://books.google.co.uk/books?id=H7ZeBAAAQBAJ&printsec=frontcover&dq=isbn:1493912771&hl=en&sa=X&ved=0ahUKEwj_rMe-xuDbAhXpA8AKHRQkB9IQ6AEIJzAA#v=onepage&q&f=false)* – Daniel Buck Jun 19 '18 at 20:23
13

Integrals containing one or more parameters are the ideal toys for doing this sort of stuff. Besides integration and differentiation w.r.t. to the parameters, analytic continuation can yield surprising results. Thing is that when we use such methods to compute a given integral, we actually engage in such a milking exercises, albeit it with the goal of reaching the given target.

E.g. I show here how to get to the result:

$$\int_{0}^{\infty}\frac{\log(x)}{x^2 -1}dx = \frac{\pi}{4}$$

Starting from the well known result:

$$\int_{0}^{\infty}\frac{x^{-p}}{1+x}dx = \frac{\pi}{\sin(\pi p)}$$

Differentiation w.r.t. to $p$ is an obvious step. Getting the minus 1 in the denominator when you start out with a denominator that is always nonzero requires introducing another parameter obtained by rescaling the integration variable, but this then needs to be analytically continued.

The nice thing about this derivation is that it avoids quite some mathematical manipulations needed to do this directly via contour integration. Moving the parameter in the complex plane, moves the pole of the integrand and if we start to push the pole against the integration contour that needed to be considered for a direct derivation, then the fact that the result is analytically continued, implies that the contour would need to be deformed to make way for the approaching pole.

So, we can avoid having to even think about choosing the right contours, taking principal values etc. etc. when manipulating integrals using parameters.

Count Iblis
  • 10,078
  • 2
  • 20
  • 43
8

Yes, especially definite integrals. To begin, I often use the function $\ln$ and its counterpart $\exp$ to start 'building' the integrand. Then I plug it in Wolfram to check that the indefinite integral doesn't have a closed form (to make sure the integral is challenging enough :)

Seeing as there is no closed form, I use substitution/by parts to create another integral. The final step is to combine the integrals at the LHS to give a constant on the right.

Some examples:

$$\small\int_0^\infty\ln\left(\frac{e^x+1}{e^x-1}\right)\,dx=2\int_0^\infty\frac{xe^x}{e^{2x}-1}\,dx=2\int_1^\infty\frac{\ln x}{x^2-1}\,dx=\frac12\int_0^\infty\frac{\ln(x+1)}{x\sqrt{x+1}}\,dx=\frac12\int_0^\infty\frac{xe^{x/2}}{e^x-1}\,dx$$

and starting with the integral $\int_0^1\frac{\ln x}{e^x}\,dx$, we find that $$\int_0^1\frac{(x^2-3x+1)\ln x}{e^x}\,dx=-\frac1e$$

The same method can be used to show that $$\int_0^\infty\frac{(x^2-3x+1)\ln x}{e^x}\,dx=0.$$

TheSimpliFire
  • 25,185
  • 9
  • 47
  • 115
  • Wow, that is very cool (though it turns out that Wolfram does give an anti derivative of the integrand of your last example). Nice! – Franklin Pezzuti Dyer Jul 20 '18 at 21:46
  • Thanks! Interesting to note that WA can't actually show the integral from 0 to infinity is zero; see [here](http://www.wolframalpha.com/input/?i=intergate+(x%5E2-3x%2B1)lnx%2Fe%5Ex) under definite integrals... – TheSimpliFire Jul 22 '18 at 09:32
5

What a delightful thread! I once spent many weeks playing with collections of line sources, free-surface pressure patches and other hydrodynamic singularities to estimate the pressures they induced on the sea-bed in finite-depth water.

Asked to give an impromptu talk to some naval engineers who wanted to know what I was working on at the time, I explained that E. O. Tuck had been using a very simple series to approximate a multiplicative term required in some integrals, and copied some notes I had to a blackboard:

$$2 \sum^{\infty}_{n=1} \frac{1}{(2n-1)^3} = \sum^{\infty}_{n=1} \frac{L_{n}}{n^{2}} = \frac{7}{4}\zeta(3)$$

where

$$L_n = 1 + \frac{1}{3} + \frac{1}{5} + \dots + \frac{1}{2n-1}.$$

The first sum is well known (23.3.20, A&S 1972, p. 807), and as there were nods of approval from the engies I thought I'd just bore them with how the 2nd series arose, so I mumbled on for a few minutes more about other work we were doing.

I remember being very impressed with their easy familiarity with the zeta function. Older now, I'm not quite so sure that I read that audience response correctly. It was a Friday afternoon, last day of term, so I suspect that some had given the free departmental sherry a bit of a nudge after our long lunch.

Lysistrata
  • 107
  • 5
5

I'm not a very experienced integrator, but I've had some fun that I'd like to share. The below are results I've reached myself, but haven't been able to prove with complete confidence.

Let $\alpha_i \in\Bbb R, \forall i\in I=\{1,2,...,n\}$ when $n\in\Bbb N$, and $\forall i,j\in I, \alpha_i=\alpha_j \iff i=j$. Then:$$\int\prod_{i\in I}(\frac{1}{x-\alpha_i})dx=C+\sum_{i\in I}\frac{\ln |x-\alpha_i|}{\prod_{i≠j\in I}(\alpha_i-\alpha_j)}$$ Via partial fraction decomposition.

clathratus
  • 15,243
  • 3
  • 14
  • 70
1

A correctly re-done version of my deleted answer.

For $a_k\ne 0$, $k=1,2,...,n$, we have $$\Pi_n:=u\left(\small{\prod_{k=1}^{n}}a_k\right)\prod_{k=1}^{n}u(a_k)=u(1)+\sum_{k=1}^{n}\sum_{\sigma\subset S_n\\ |\sigma|=k}u\left(\small{\prod_{j\in\sigma}}a_j^2\right),\tag{1}$$ where $S_n=\{1,2,...,n\}$ (from here), with $u(x)=x+1/x$. Letting $a_k=\exp(i\theta_k)$, we have $u(a_k)=2\cos\theta_k$ and $$u\left(\small{\prod_{j\in\sigma}}a_j^2\right)=u\left(\small{\prod_{j\in\sigma}}e^{2i\theta_j}\right)=u\left(e^{2i\small{\sum_{j\in\sigma}}\theta_j}\right)=2\cos\left(2\small{\sum_{j\in\sigma}}\theta_j\right).$$ Thus $$\Pi_n=2^{n+1}\cos\left(\small{\sum_{k=1}^n\theta_k}\right)\prod_{k=1}^{n}\cos\theta_k=2+2\sum_{k=1}^{n}\sum_{\sigma\subset S_n\\ |\sigma|=k}\cos\left(2\small{\sum_{j\in\sigma}}\theta_j\right).$$ Setting $\theta_k=x$, $$\Pi_n=2^n\cos(nx)\cos(x)^n=1+\sum_{k=1}^{n}\sum_{\sigma\subset S_n\\ |\sigma|=k}\cos(2|\sigma|x)=1+\sum_{k=1}^{n}{n\choose k}\cos(2kx),$$ because $${n\choose k}=\left|\left\{\sigma\subset S_n:|\sigma|=k\right\}\right|.$$ Thus $$\int_0^{a}\cos(nx)\cos(x)^ndx=\frac1{2^n}\left(a+\sum_{k=1}^{n}{n\choose k}\frac{\sin(2ka)}{2k}\right),$$ which, I don't doubt, can be used to develop more interesting identities.


An identity developed from the above:

$$\int_0^1t\sin\left(n(n+1)t\right)\cos(nt)^{n-1}dt=\frac1{2^n n^2}\sum_{k=1}^{n}{n\choose k}\left(\frac{\sin(2nk)}{2nk}-\cos(2nk)\right),$$ where $n\in\Bbb Z_{\ge1}$.

clathratus
  • 15,243
  • 3
  • 14
  • 70