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Question
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
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Solution
Let `I = int_0^(pi/2) sqrtsinx/(sqrt sinx + sqrt cos x) dx` ...(i)
Replace x to `(pi/2 - x)` in (i)
`[∵ 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`
`I = int_0^(pi/2) sqrtcosx/(sqrtcos x + sqrt sin x) dx` ...(ii)
Adding (i) and (ii), we get
`2I = int_0^(pi/2) [sqrt sinx/ (sqrt sinx + sqrt cos x) + sqrt cos x/(sqrt cos x + sqrt sinx)] dx`
`= int_0^(pi/2) (sqrt cos x + sqrt sin x)/(sqrt cosx + sqrt sin x)`
`= int_0^(pi/2) dx = [x]_0^(pi/2)`
`= pi/2 - 0`
`= pi/2`
⇒ `I = pi/4`
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