By using the properties of definite integrals, evaluate the integrals `int0a(sqrt(x))/(sqrt(x)+sqrt(a-x))dx`

Explanation

`" Let I "= int_(0)^(a) (sqrt(x))/(sqrt(x)+sqrt(a-x))dx " "....(1)`
`rArr Iint_(0)^(a) (sqrt(a-x))/(sqrt(a-x)+sqrt(a-(a-x)))dx`
`[ :' int_(0)^(a) f(x) dx= int_(0)^(a) f(a-x)dx]`
`=int_(0)^(a) (sqrt(a-x))/(sqrt(a-x)+sqrt(x))dx" "......(2)`
Adding equation (1) and (2)
`2I =int_(0)^(a) (sqrt(x)+sqrt(a-x))/(sqrt(x)+sqrt(a-x))dx= int_(0)^(a) 1 dx =[x]_(0)^(a)=a`
`rArr I =(a)/(2)`