Question

Using properties of definite integrals, evaluate 

`int_0^(π/2)  sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`

Answer 1

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)`