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Evaluate: $$\int_0^{\pi} x\csc^{\sin(\cos x)}(x)\,dx$$

I honestly don't know how to deal with this case. If I apply the property $\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$, I get: $$\int_0^{\pi} (\pi-x) \frac{1}{(\sin x)^{-\sin(\cos x)}}\,dx$$ but I don't think this is going to help.

Any help is appreciated. Thanks!

Pranav Arora
  • 11,315
  • This seems like a tough integral. Where does this come from? – John M May 29 '14 at 18:20
  • I got it from one of the problem solving website and this problem was a user-shared problem. Its been months and there isn't a single solution to this problem yet. I finally decided to post it on Math.SE. I doubt this has got a nice solution but since I am no expert, I would like to know the opinion of members here. – Pranav Arora May 29 '14 at 18:22
  • The second line should be $$ \int_0^{\pi} (\pi-x)\csc^{-\sin(\cos x)}(x),dx. $$ It's still difficult anyway. :P – Tunk-Fey May 29 '14 at 18:24
  • Woops! I will edit that. :D – Pranav Arora May 29 '14 at 18:24
  • The integrand function is not well defined for real numbers $x$, as the base function takes on negative values in $[0, \pi]$ and the exponent takes on non-integer values. – Sammy Black May 29 '14 at 18:28
  • @SammyBlack: I am not sure if I understood you. Do you mean $\csc (x)$ takes negative values in $[0,\pi]$? I doubt it. – Pranav Arora May 29 '14 at 18:30
  • You can rewrite the exponential expression to see this more clearly: $x \csc^{\sin( \cos x )}(x) = x \left[ \csc(x) \right]^{\sin( \cos x )} = x \exp \left[ \sin( \cos x ) \ln ( \csc x ) \right]$. – Sammy Black May 29 '14 at 18:30
  • Oh! Of course. My mistake. Hmmm. – Sammy Black May 29 '14 at 18:31
  • In searching for a symmetry (which might make the integral identically $0$), I stumbled upon the functional equation $f(x) \cdot f(\pi - x) = x \cdot (\pi - x)$, but I can't find much use for it. – Sammy Black May 29 '14 at 18:35
  • Yes, it isn't zero. W|A gives 3.99052. You can look at some of the possible closed forms to this numerical answer here: http://www.wolframalpha.com/input/?i=3.99052 . – Pranav Arora May 29 '14 at 18:38
  • I dropped a high precision result into the Inverse Symbolic Calculator at http://isc.carma.newcastle.edu.au and came up with nothing. – John M May 29 '14 at 18:49
  • Duplicate of http://math.stackexchange.com/questions/1032284/evaluating-int-0-pi-fracx-sin-x-sin-cos-xdx – Jack D'Aurizio Mar 09 '17 at 16:16
  • Actually this question was posted before the linked one. It is active now. – Тyma Gaidash Jun 26 '21 at 13:11

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