Integrate the following functions with respect to x: (6-x)(x-4) - Mathematics

Integrate the following functions with respect to x:

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

Sum

`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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Chapter 11: Integral Calculus - Exercise 11.12 [Page 225]

Chapter 11 Integral Calculus
Exercise 11.12 | Q 1. (iii) | Page 225

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