Question

Evaluate the following:

`int sqrt(2"a"x - x^2)  "d"x`

Answer 1

Let I = `int sqrt(2"a"x - x^2)  "d"x`

= `int sqrt(-(x^2 - 2"a"x))  "d"x`

= `int sqrt(-(x^2 - 2"a"x + "a"^2 - "a"^2))  "d"x`

= `int sqrt(-[(x - "a")^2 - "a"^2])  "d"x`

= `int sqrt("a"^2 - (x - "a")^2)  "d"x`

= `(x - "a")/2 sqrt("a"^2 - x^2) + "a"^2/2  sin^-1  ((x - "a")/"a") + "C"`  ......`[because int sqrt("a"^2 - x^2) "d"x = x/2sqrt("a"^2 - x^2) - "a"^2/2  sin^-1  x/"a" + "C"]`

= `(x - "a")/2 sqrt("a"^2 - (x^2 - 2"a"x + "a"^2)) + "a"^2/2  sin^-1  ((x - "a")/"a") + "C"`

= `(x - "a")/2 sqrt(2"a"x - x^2) + "a"^2/2 sin^-1  9(x - "a"0/"a") + "C"`

Hence, I = `(x - "a")/2 sqrt(2"a"x - x^2) + "a"^2/2 sin^-1  ((x - "a")/"a") + "C"`.