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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 arjafi Dec 31 '15 at 15:18

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

Now let's just hope that your father doesn't read – Tim Pietzcker Jul 9 '12 at 6:34
Wiki-hammered as this is a bit list with a subjective (as hard as possible) answer. – Willie Wong Jul 9 '12 at 8:33
Yeah... or your dad might know right away that the solution evaluates to 59... on his birthday... when he just turned 59. – phatfingers Jul 10 '12 at 18:29
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. – Mlazhinka Shung Gronzalez LeWy Dec 19 '13 at 18:12
up vote 81 down vote accepted

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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Is that $$\ln(x^2)$$ or $$(\ln x)^2$$ ? – ypercubeᵀᴹ Jul 9 '12 at 12:06
@ypercube: the second one. – J. M. Jul 9 '12 at 12:43
Added to the card. Thanks very much! – Argon Jul 11 '12 at 1:11
How does one start to solve this? – yiyi Dec 11 '12 at 8:50
Using the Risch-Norman algorithm, the antiderivative is not elementary. Maple doesn't find a closed-form antiderivative. – Robert Israel Oct 30 '13 at 14:31

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

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With added benefit of having very few arbitrary-looking constants. – Alex Feinman Jul 9 '12 at 18:27
@Alex: should be even better in 2 years! :-) – Raymond Manzoni Jul 9 '12 at 18:37
I like this Easy to read, no crazy exponents yet nontrivial solution. – Chad Jul 9 '12 at 19:12
@Joe: hint for fast evaluation : use the generating function for Euler numbers. – Raymond Manzoni Mar 15 '13 at 12:18
@Raymond 59 = 61 - 2? – Joe Z. Mar 15 '13 at 12:29

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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@PeterTamaroff Thanks. – Alex Becker Jul 9 '12 at 3:33
Out of curiosity, why the downvote? – Alex Becker Jul 9 '12 at 6:29
@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? – Jakob S. Jul 9 '12 at 11:23
@JoachimSauer That was actually a typo on my part. $108$ was meant to be $118$. – Alex Becker Jul 11 '12 at 3:07
@حكيمالفيلسوفالضائع It's only a conjecture? – Soke May 2 '14 at 21:36

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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Waiting on your edit as promised.... :-) – msh210 Jun 13 '13 at 5:16
Ah, shoot; let me look for my notes on this... if memory serves, I trolled through OEIS and looked for generating functions. – J. M. Jun 13 '13 at 5:24
7 – wchargin May 12 '14 at 0:09

There's also

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

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The 118 is a bit too obvious, assuming it's not just coincidence that 118/2 = 59 – Random832 Jul 9 '12 at 21:13
@Random832 True, but I figured I'd write down an integral that gives his father a reasonable chance of figuring it out... least I myself would struggle with some of the other alternatives. – user12014 Jul 9 '12 at 21:17

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