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Question
Using properties of definite integrals, evaluate
`int_0^(π/2) sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`
Solution
I = `int_0^(π/2) sqrt(sin x )/ (sqrtsin x + sqrtcos x) dx` ...(i)
= `int_0^(π/2) sqrtsin (π/2 - x)/ (sqrtsin (π/2 - x) + sqrtcos (π/2 - x) dx`
by using `int_0^a f (x) dx = int_0^a f (a - x ) dx`
I = `int_0^(π/2) sqrt(cos x )/ (sqrtcos x + sqrtsin x) dx` ...(ii)
Adding equations (i) and (ii), we have
2I = `int_0^(π/2) (sqrtsin x + sqrtcos x )/ (sqrtsin x + sqrtcos x) dx`
2I = `int_0^(π/2) 1 dx = [x]_0^(π/2)`
I = `(1)/(2) [ (π)/(2) - 0 ]`
I = `(π)/(4)`
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