Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x.$$
I tried using by parts and complex numbers along with series expansion but I was unable to find the answer.
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5Why would this have a closed form? The limits are not very natural for the integrand. – user111187 Nov 21 '14 at 11:40
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@user111187 - I assume the limits are to ensure the integral is defined, as if $\sin x$ is negative, the denominator is undefined. – Nathaniel Bubis Nov 21 '14 at 11:42
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4Finding a closed form for the antiderivative seems (at least to me) to be a dream. – Claude Leibovici Nov 21 '14 at 11:46
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4Just for hee-haws, it turns out that $$\frac{\cos{x}}{(\sin{x})^{\sin{\cos{x}}}}$$ has an antiderivative in closed form. – Ron Gordon Nov 21 '14 at 11:47
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2@RonGordon. Please, tell more and show us. I stop my dream and I am awake. How did you get an answer ? – Claude Leibovici Nov 21 '14 at 11:52
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1@ClaudeLeibovici: I messed around in Mathematica. Just guessing around, that's all. I was as surprised as you. The result is $$\frac{\sin ^{1-\sin (\cos (x))}(x)}{1-\sin (\cos (x))}$$ – Ron Gordon Nov 21 '14 at 11:53
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@RonGordon. Moreover, it is so simple expression ! How did you think about $\cos(x)$ ? – Claude Leibovici Nov 21 '14 at 11:55
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6@ClaudeLeibovici: I spend my days just thinking about cosines. Every problem, I ask: How can I put a cosine into it? – Ron Gordon Nov 21 '14 at 11:56
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2@ClaudeLeibovici: OK, bad news...I checked the result in Mathematica and it does not return the original integrand as far as I can tell. I maybe uncovered a Mathematica bug. More proof to not trust a machine. – Ron Gordon Nov 21 '14 at 12:01
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@RonGordon. Using your result, the integral should be $0$ while numerical integration gives $5.66895$ ! Good catch and enjoy with cosine ! Cheers :-) – Claude Leibovici Nov 21 '14 at 12:13
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1@ClaudeLeibovici: yes, I should take my own advice and check Mathematica results before posting. Sigh – Ron Gordon Nov 21 '14 at 12:16
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4The one equals $$3.9905179137710221001990755970868667785510732535206. $$ – user64494 Nov 21 '14 at 12:40
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Do you mean in radians? – Math Lover Jan 10 '19 at 13:44
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I think the “quotient rule” will help. https://math.stackexchange.com/questions/68505/is-there-a-rule-of-integration-that-corresponds-to-the-quotient-rule – Math Lover Jan 10 '19 at 13:46
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This function has a part that can be written as $f(x)^{g(x)}$, so I'm pretty sure there isn't an antiderivative in closed form, as @ClaudeLeibovici said. – Zuter_242 May 20 '21 at 18:56
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maybe $u = \pi - x$ – hellofriends Jun 26 '21 at 01:31
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Here is a graph which explains the bounds. This almost looks like the integral here, but yours is not even or odd. – Тyma Gaidash Jun 26 '21 at 12:02