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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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closed as too broad by user642796 Dec 31 '15 at 15:18

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Now let's just hope that your father doesn't read math.stackexchange.com... $\endgroup$ – Tim Pietzcker Jul 9 '12 at 6:34
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    $\begingroup$ Wiki-hammered as this is a bit list with a subjective (as hard as possible) answer. $\endgroup$ – Willie Wong Jul 9 '12 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. $\endgroup$ – phatfingers Jul 10 '12 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 '13 at 18:12
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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 $$(\ln x)^2$$ ? $\endgroup$ – ypercubeᵀᴹ Jul 9 '12 at 12:06
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    $\begingroup$ @ypercube: the second one. $\endgroup$ – J. M. is a poor mathematician Jul 9 '12 at 12:43
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    $\begingroup$ Added to the card. Thanks very much! $\endgroup$ – Argon Jul 11 '12 at 1:11
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    $\begingroup$ How does one start to solve this? $\endgroup$ – yiyi Dec 11 '12 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. $\endgroup$ – Robert Israel Oct 30 '13 at 14:31
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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. $\endgroup$ – Alex Feinman Jul 9 '12 at 18:27
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    $\begingroup$ I like this Easy to read, no crazy exponents yet nontrivial solution. $\endgroup$ – Chad Jul 9 '12 at 19:12
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    $\begingroup$ Wolfram Alpha timed out trying to evaluate it. $\endgroup$ – Joe Z. Mar 14 '13 at 14:57
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    $\begingroup$ @Joe: hint for fast evaluation : use the generating function for Euler numbers. $\endgroup$ – Raymond Manzoni Mar 15 '13 at 12:18
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    $\begingroup$ @Raymond 59 = 61 - 2? $\endgroup$ – Joe Z. Mar 15 '13 at 12:29
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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. $\endgroup$ – Alex Becker Jul 9 '12 at 3:33
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    $\begingroup$ Out of curiosity, why the downvote? $\endgroup$ – Alex Becker Jul 9 '12 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 '12 at 11:23
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    $\begingroup$ @JoachimSauer That was actually a typo on my part. $108$ was meant to be $118$. $\endgroup$ – Alex Becker Jul 11 '12 at 3:07
  • $\begingroup$ @حكيمالفيلسوفالضائع It's only a conjecture? $\endgroup$ – MCT May 2 '14 at 21:36
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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 '13 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. $\endgroup$ – J. M. is a poor mathematician Jun 13 '13 at 5:24
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    $\begingroup$ i.imgur.com/8PfsQ.png $\endgroup$ – wchargin May 12 '14 at 0:09
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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 '12 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 '12 at 21:17

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