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`int_0^(pi/2)(sqrt(sinx))/(sqrt(sinx)+sqrt(cosx)) dx`

Text Solution

Verified by Experts

Let `I = int_(0)^(pi/2) (sqrt(sinx))/((sqrt(sinx)+sqrt(cosx))) dx ->(1)`
`int_0^a f(x) dx = int_0^a f(a-x) dx`
`:. I = int_(0)^(pi/2) (sqrt(sin(pi/2-x)))/((sqrt(sin(pi/2-x))+sqrt(cos(pi/2-x)))) dx`
`=>int_(0)^(pi/2) (sqrt(cosx))/((sqrt(cosx)+sqrt(sinx))) dx ->(2)`
Now, adding (1) and (2),
`2I = int_(0)^(pi/2) ((sqrt(sinx))/((sqrt(sinx)+sqrt(cosx))) + (sqrt(cosx))/((sqrt(cosx)+sqrt(sinx))) ) dx`
`=>2I = int_(0)^(pi/2) dx`
`=>2I = [x]_0^(pi/2)`
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