# The output of this integral seems irrational, but how would you go about...

The output of this integral seems irrational, but how would you go about proving it? I can't even integrate it without using computer assisted estimations

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The output of this integral seems irrational, but how would you go about proving it?

Hard to do that when you can't even represent the integral exactly in reasonable terms.

Integral equals some integers p/q but at a certain point you need to concretely evaluate the integral, which is resisting all of my efforts

I don't really give a fuck about pure math so I always approx my integrals with a computer. What do mathematicians even do when then run into a function that has no closed-form anti-derivative?

No one cares about exact values for integrals. Analysts are usually happy with just knowing it's finite.

The output of this integral seems irrational, but how would you go about proving it?
I don't know, but I guess that the proof isn't simple, at least on the first glance.
Maybe you had a better chance if you could express it in its closed form?

Usually you'd do it by first finding a bounds for the solution (let's call it I) that's between two fractions. Then you'd show that for every such interval there's a smaller interval between two fractions for where I lies, so I cannot be a fraction.

For example I is in (a/b, c/d) => I is in (a2/b2, c2/d2), where a2/c2 > a/c and c2/d2 < c/d.
It's been a long time since I done this, but I remember proving irrationality for integrals being somewhat easy for first year... I could be completely wrong. Good luck

Estimate it using a series expansion, then show it is close to a rational (within 1/q^2 of p/q)

Taylor series. Approximate the function with a polynomial and then calc the area of the polynomial. Approximation gets better and better with more terms.

It's probably impossible. Just look at how unnatural that integrand is.

How about setting z:=exp(ix) and play around with complex integrals?

compute infinity many digits then check for repetition

This is actually really interesting.
Have you tried evaluation using one of the sine integral functions? Or simply naming a function to be its integral?
Something like
[Math]\int x^{cos(x)} = \phi(x) [\math] and then based on that you analyse its properties.
Then a proof by contradiction could work but im not a mathematitian.

Shit. I placed the wrong slash.

[Math]\int x^{cos(x)} = \phi(x) [/math]

You wrote "Math" instead of "math".
$\int x^{cos(x)} = \phi(x)$

All I can say is that the function is tough to work with and the expansion converges amazingly badly

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Ill off myself its no problem

Approximation gets better and better with more terms.
Only if the function is analytic

Why do you even care? -Physicist

it's irrational, you can tell easily from the series

it's not an algebraic antiderivative so its automatically transcendental this irrational (basic algebra)

it's not an algebraic antiderivative so its automatically transcendental this irrational (basic algebra)
elaborate

it's irrational, you can tell easily from the series
elaborate

Actually beginning from the very next decimal place it repeats itself.