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Question
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Solution
Let I = `int(5 - 3x)(2 - 3x)^(-1/2).dx`
Put 2 – 3x = t
∴ –3dx = dt
∴ dx = `(-dt)/(3)`
Also, x = `(2 - t)/(3)`
∴ I = `int[5 - 3((2 - t)/3)]t^(-1/2).((-dt)/(3))`
= `-1/3(5 - 2 + t)t^(-1/2)dt`
= `-1/3 int(3 + t)t^(-1/2) dt`
= `-1/3 int(3t^(-1/2) + t^(1/2))dt`
= `-3/3 int t^(-1/2)dt - (1)/(3) int t^(1/2) dt`
= `-t^(1/2)/((1/2)) - (1)/(3).t^(3/2)/((3/2)) + c`
= `-2sqrt(2 - 3x) - (2)/(9)(2 - 3x)^(3/2) + c`
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