all your integrals are flying away ⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚⨙⨚

>>8778549
Well the integrals shown are closed, so they will come back eventually.

>>8778551
>Well the integrals shown are closed, so they will come back eventually.
how does that conclusion follow?

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>>8778551

>>8778611
by the definition of a closed interval

>>8778551
[math]...^{-1^{-1^{-1^{-1^{...}}}}}[/math]

[math]...{-1^{-1^{-1^{-1^{...}}}}}[/math]

[math]..._{-1^{-1^{-1^{-1^{-1^{-1}}}}}}[/math]

>>8778639
what. the. fuck.

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if one integral flies away, followed by two more integrals, followed by three more integrals, followed by four, etc. etc., eventually -1/12 integrals will have gone away. so if I wait for that to happen twelve times, one integral will have come back. from there it's just simply repeating the process for each other integral.
Q.E.D.

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This board is loaded with brainlets, so I'd say 90% of the people on here can't do this simple problem.

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>>8778699

>>8779116
easy for [math]a\in\mathbb{N}[/math]. Now try [math]a\in\mathbb{R}[/math]

>>8779957
substitute u^(1/a)=x

>>8779968
with that sub:
[eqn]\int x^a \sin x \mathrm{d}x=\int \frac{1}{a}u^{1/a}\sin u^{1/a}\mathrm{d}u[/eqn]
congrats, you just made the integral harder

>>8779990
then you substitute v=u^1/a and you have integral of v*sinv

>>8780000
kek

>>8778549
0>1

(My logic is some of them may return to fix the mess I just made, kek)

>>8779990
>>8779957

Computing a few of them for fun makes it look like there is no general algorithm or solution for them. If you try simple rationals like 2.5 it becomes a rational polynumber with some trig functions. If you put in irrational numbers like square root of 2 or pi then it needs the gamma function.

I think if you are a calculus/analysis guy then you could at best prove that the integral always exists and if you are an algebra guy you could maybe outline a procedure for rational numbers and irrational numbers using non-elementary functions. Seems hard though.

>>8778549
Making them fly around a loop lol

>>8778699
>>8779990
cant be solved in terms of elementary functions, you need to use the incomplete gamma function im pretty sure