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My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$.

I want to try and make a definite integral that equals $59$. So far I can only think of ones that are easy to evaluate. I was wondering if anyone had a definite integral (preferably with no elementary antiderivative) that is difficult to evaluate and equals $59$? Make it as hard as possible, feel free to add whatever you want to it!

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    $\begingroup$ Now let's just hope that your father doesn't read math.stackexchange.com... $\endgroup$ Jul 9, 2012 at 6:34
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    $\begingroup$ Wiki-hammered as this is a bit list with a subjective (as hard as possible) answer. $\endgroup$ Jul 9, 2012 at 8:33
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    $\begingroup$ Yeah... or your dad might know right away that the solution evaluates to 59... on his birthday... when he just turned 59. $\endgroup$ Jul 10, 2012 at 18:29
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    $\begingroup$ The fun is on proving it. So, it doesn't matter if the father sees this post or if he/she knows the answer is 59. $\endgroup$
    – OR.
    Dec 19, 2013 at 18:12

5 Answers 5

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compact : $$\int_0^\infty \frac{(x^4-2)x^2}{\cosh(x\frac{\pi}2)}\,dx$$

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    $\begingroup$ With added benefit of having very few arbitrary-looking constants. $\endgroup$ Jul 9, 2012 at 18:27
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    $\begingroup$ @Alex: should be even better in 2 years! :-) $\endgroup$ Jul 9, 2012 at 18:37
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    $\begingroup$ I like this Easy to read, no crazy exponents yet nontrivial solution. $\endgroup$
    – Chad
    Jul 9, 2012 at 19:12
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    $\begingroup$ Wolfram Alpha timed out trying to evaluate it. $\endgroup$
    – Joe Z.
    Mar 14, 2013 at 14:57
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    $\begingroup$ @Raymond 59 = 61 - 2? $\endgroup$
    – Joe Z.
    Mar 15, 2013 at 12:29
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You might try the following: $$ \frac{64}{\pi^3} \int_0^\infty \frac{ (\ln x)^2 (15-2x)}{(x^4+1)(x^2+1)}\ dx $$

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    $\begingroup$ Is that $$\ln(x^2)$$ or $$(\ln x)^2$$ ? $\endgroup$ Jul 9, 2012 at 12:06
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    $\begingroup$ @ypercube: the second one. $\endgroup$ Jul 9, 2012 at 12:43
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    $\begingroup$ Added to the card. Thanks very much! $\endgroup$
    – Argon
    Jul 11, 2012 at 1:11
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    $\begingroup$ How does one start to solve this? $\endgroup$
    – yiyi
    Dec 11, 2012 at 8:50
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    $\begingroup$ Using the Risch-Norman algorithm, the antiderivative is not elementary. Maple doesn't find a closed-form antiderivative. $\endgroup$ Oct 30, 2013 at 14:31
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Combining an very difficult infinite sum with the indefinite integral of $\sin(x)/x$ over $\mathbb R$, which has no elementary antiderivative, gives

$$\frac{118\sqrt{2}}{9801}\int_{\mathbb R} \left(\sum_{k=0}^\infty \left(\frac{(4k)!(1103+26390k)}{(k!)^4396^{4k}}\frac{\sin x}{x}\right)\right)dx=59\cdot \frac{1}{\pi}\cdot \pi=59$$

which should be tough enough to stump anyone who hasn't seen them before.

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  • $\begingroup$ @PeterTamaroff Thanks. $\endgroup$ Jul 9, 2012 at 3:33
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    $\begingroup$ Out of curiosity, why the downvote? $\endgroup$ Jul 9, 2012 at 6:29
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    $\begingroup$ @JoachimSauer: It's probably due to my dumbness, but I don't see where the 59 is hidden in 108? 59*2=118, if you meant that? $\endgroup$
    – Jakob S.
    Jul 9, 2012 at 11:23
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    $\begingroup$ @JoachimSauer That was actually a typo on my part. $108$ was meant to be $118$. $\endgroup$ Jul 11, 2012 at 3:07
  • $\begingroup$ @حكيمالفيلسوفالضائع It's only a conjecture? $\endgroup$
    – MT_
    May 2, 2014 at 21:36
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There's also

$$\int_0^\infty \! x^3 e^{-(118)^{-1/2}x^2} \, dx$$

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    $\begingroup$ The 118 is a bit too obvious, assuming it's not just coincidence that 118/2 = 59 $\endgroup$
    – Random832
    Jul 9, 2012 at 21:13
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    $\begingroup$ @Random832 True, but I figured I'd write down an integral that gives his father a reasonable chance of figuring it out... ...at least I myself would struggle with some of the other alternatives. $\endgroup$
    – user12014
    Jul 9, 2012 at 21:17
  • $\begingroup$ Yeah! Our objective is not to teach him mathematics for that instance rather to provide him with the pleasure, challenge and thrill of solving such integral!!! Isn't it.... $\endgroup$ Jan 12, 2021 at 4:43
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Somewhat complicated, but...

$$\begin{align*}\frac{12}{\pi}\int_0^{2\pi} \frac{e^{\frac12\cos\,t}}{5-4\cos\,t}&\left(2\cos \left(t-\frac{\sin\,t}{2}\right)+3\cos\left(2t-\frac{\sin\,t}{2}\right)+\right.\\&\left.14\cos\left(3t-\frac{\sin\,t}{2}\right)-8\cos\left(4t-\frac{\sin\,t}{2}\right)\right)\mathrm dt=59\end{align*}$$

As a hint on how I obtained this integral, I used Cauchy's differentiation formula on a certain function (I'll edit this answer later to reveal that function), and took the real part...

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    $\begingroup$ Waiting on your edit as promised.... :-) $\endgroup$
    – msh210
    Jun 13, 2013 at 5:16
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    $\begingroup$ Ah, shoot; let me look for my notes on this... if memory serves, I trolled through OEIS and looked for generating functions. $\endgroup$ Jun 13, 2013 at 5:24
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    $\begingroup$ i.imgur.com/8PfsQ.png $\endgroup$
    – wchargin
    May 12, 2014 at 0:09
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    $\begingroup$ Almost 8 years later. You still haven't posted the solution. Please... $\endgroup$ Oct 10, 2021 at 2:15
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    $\begingroup$ Still waiting for the solution! :D $\endgroup$
    – Max0815
    Jan 22, 2023 at 2:01

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