By using properties of definite integrals, evaluate the integrals
$\int_{0}^{\frac{π}{2}} \frac{\sqrt{s i n} x}{\sqrt{s i n x} + \sqrt{c o s x}} d x$

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Solution

Let I= $\int_{0}^{\frac{π}{2}} \frac{\sqrt{s i n} x}{\sqrt{s i n} x + \sqrt{c o s} x} d x$

Then $I = \int_{0}^{\frac{π}{2}} \frac{\sqrt{s i n (\frac{π}{2}) - x}}{s i n \sqrt{(\frac{π}{2} - x)} + \sqrt{c o s (\frac{π}{2} - x)}} d x \Rightarrow I = \int_{0}^{\frac{π}{2}} \frac{\sqrt{c o s x}}{\sqrt{c o s x} + \sqrt{s i n x}} d x [∵ s i n (\frac{π}{2} - x) = c o s x a n d c o s (\frac{π}{2} - x) = s i n x]$
On adding Eqs. (i) and (ii) , we get $2 I = \int_{0}^{\frac{π}{2}} \frac{\sqrt{s i n x} + \sqrt{c o s x}}{\sqrt{s i n x} + \sqrt{c o s x}} d x = \int_{0}^{\frac{π}{2}} 1 d x = [x]_{0}^{\frac{π}{2}} = \frac{π}{2} - 0 \Rightarrow 1 = \frac{π}{4}$

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