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Is there any closed formula for $$ \int_{0}^{\pi/2} \dfrac{e^{-x}\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}} $$ I know $$ \int_{0}^{\pi/2} \dfrac{\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}} = \dfrac{\pi}{4} $$ replacing $x$ by $\pi/2 - y$. Thanks for any help.

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There exists no closed form for your proposed Integral. However if we generalize the problem as $$I(\alpha ) = \int\limits_0^{\frac{\pi }{2}} {\frac{{{e^{ - \alpha x}}\sqrt {\cos x} }}{{\sqrt {\cos x} + \sqrt {\sin x} }}dx}$$ we could obtain the variation of the integral with respect to $\alpha$ (numerically) asenter image description here

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    $\begingroup$ How do you know there is none? $\endgroup$ – nbubis Jul 20 '14 at 4:21

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