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What is definite integral `int_0^(pi/2) ` (1 - sin x)^(1/2)dx?

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What is definite integral `int_0^(pi/2) ` (1 - sin x)^(1/2)dx?

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`1-sin(x)=sin^2(x/2)+cos^2(x/2)-2sin(x/2)cos(x/2)`

`=(cos(x/2)-sin(x/2))^2`

`` Thus

`(1-sin(x))^(1/2)=cos(x/2)-sin(x/2))`

`int_0^(pi/2)(1-sin(x))^(1/2)dx=int_0^(pi/2)(cos(x/2)-sin(x/2))dx`

`=2sin(x/2)+2cos(x/2)}_0^(pi/2)`

`=2{(1/sqrt(2)+1/sqrt(2))-(0+1)}`

`=2(sqrt(2)-1)`

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