Integral involving square root of sine and cosine 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.
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) as
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) as