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Can anyone help me in evaluating the following integral : $\int_{0}^{\pi}\frac{\sin (x+\sqrt x)}{\cos(x-\sqrt x)} dx$. I tried doing this by substitution but didn't work. Also by parts looks cumbersome.

creative
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  • It looks horrific, try contour integration to avoid actually evaluating the integral. – orion Aug 12 '14 at 07:15
  • I agree with orion. Did you encounter this integral somewhere? Do you have any reason to believe the integral is doable? – David H Aug 12 '14 at 07:22
  • As said in previous comments, evaluating the integral looks to be a potential nightmare. If contour integration does not work, use numerical methods. – Claude Leibovici Aug 12 '14 at 08:40
  • I found this in a paper while I was searching for something else in my study cupboard. I don't know whether it exists or not. But nevertheless I tried various methods but none worked. May be I should try it numerically. – creative Aug 12 '14 at 08:48
  • Have you tried expanding the numerator and denominator using angle addition formulas for the sine and cosine function, and then employing Fresnel integrals? – Lucian Aug 12 '14 at 10:21
  • I don't think this converges on $(0, \pi)$. It's a $\tan$ function on it's period... – m0nhawk Aug 12 '14 at 14:28
  • Wolframalpha gives the value $ -0.851945$. So apparently it converges. The number of digits is too short to guess on the exact number (ISC gives a lot of possibilities) – Yuriy S Feb 25 '16 at 14:13
  • Mathematica gives $-0.851945423233506$. No result from ISC or Wolframalpha – Yuriy S Apr 07 '16 at 17:54

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