Evaluate the following :
`int_(1)^(2)(sqrtx)/(sqrt(3-x)+sqrtx)dx.`

Explanation

Let `I=int_(1)^(2)(sqrtx)/(sqrt(3-x)+sqrtx)dx`
We use the property, `int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx`.
Hence in I, we change x by `1+2-x.`
`therefore I=int_(1)^(2)(sqrt(1+2-x))/(sqrt(3-1-2+x)+sqrt(1+2-x))dx`
`=int_(1)^(2)(sqrt(3-x))/(sqrtx+sqrt(3-x))dx" ...(2)"`
Adding (1) and (2), we get,
`2I=int_(1)^(2)(sqrtx)/(sqrt(3-x)+sqrtx)dx+int_(1)^(2)(sqrt(3-x))/(sqrtx+sqrt(3-x))dx`
`=int_(1)^(2)(sqrtx+sqrt(3-x))/(sqrt(3-x)+sqrtx)dx`
`=int_(1)^(2)1dx=[x]_(1)^(2)=2-1=1`
`therefore I=(1)/(2).`