The value of Definite Integral $\displaystyle \displaystyle \int_{0}^{\frac{\pi}{2}}(\sin x+\cos x)\sqrt{\frac{e^x}{\sin x}}dx$
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Split up the integral $\displaystyle \int_0^{\pi/2} e^{x/2} \sqrt{\sin x} dx + \int_0^{\pi/2} e^{x/2} \dfrac{\cos x}{\sqrt{\sin x }}dx$ Use integration by parts on the second (choose $\displaystyle u = e^{x/2} $) and see what happens.
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