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Question
Integrate the following functions with respect to x:
`sqrt((6 - x)(x - 4))`
Solution
`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"`
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