# Existence of the integral $\int_{0}^{\pi}\frac{\sin (x+\sqrt x)}{\cos(x-\sqrt x)} dx$

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.

• It looks horrific, try contour integration to avoid actually evaluating the integral. – orion Aug 12 '14 at 7:15
• 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 7:22
• 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 8:40
• 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 8:48
• 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