Question

Evaluate:

`int_0^5 sqrt(x)/(sqrt(5 - x) + sqrt(x)) dx`

Answer 1

I = `int_0^5 sqrt(x)/(sqrt(5 - x) + sqrt(x)) dx`

Using the property,

`int_0^a f(x) = int_0^a f(a - x) dx`

`I = int_0^5 (sqrt(5 - x))/(sqrt(x) + sqrt(5 - x)) dx`

`2I = int_0^5 (sqrt(5 - x)  dx)/(sqrt(5 - x) + sqrt(x)) + int_0^5 (sqrt(x)  dx)/(sqrt(5 - x) + sqrt(x))`

`2I = int_0^5 1dx = [x]_0^5`

`I = 5/2`

I = 2.5