Question

Integrate the following functions with respect to x:

`sqrt((6 - x)(x - 4))`

Answer 1

`int sqrt((6 - x)(x - 4))  "d"x = int sqrt(6x - 24 - x^2 + 4x)  "d"x`

= `int sqrt(10x - x^2 - 24)  "d"x`

= `int sqrt(- 24 - (x^2 - 10x))  "d"x`

= `int sqrt(- 24 - [(x - 5)^2 - 5^2])  "d"x`

= `int sqrt(- 24 - (x - 5)^2 + 25)  "d"x`

= `int sqrt(1 - (x - 5)^2  "d"x`

Put x – 5 = 1

dx = dt

= `int sqrt(1^2 - "t"^2)  "dt"`

= `"t"/2 sqrt(1^2 - "t"^2) + 1^2/2 sin^-1 ("t"/1) + "c"`

= `(x - 5)/2 sqrt(1 - (x - 5)^2) + 1/2 sin^-1 ((x - 5)/1) + "c"`

= `(x - 5)/2 sqrt(1 - (x^2 - 10x + 25)) + 1/2 sin^-1 (x - 5) + "c"`

= `(x - 5)/2 sqrt(1 - x^2 + 10x - 25) + 1/2 sin^-1 (x - 5) + "c"`

`int sqrt((6 - x)(x - 4))  "d"x = (x - 5)/2 sqrt(10x - x^2 - 24) + 1/2 sin^-1 (x - 5) + "c"`