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Question
Integrate the function:
`sqrt(1+ 3x - x^2)`
Solution
Let `I = int sqrt (1 + 3x - x^2) dx`
`= int sqrt (1 - (x^2 - 3x)) dx`
`= int sqrt (1 - (x^2 - 3x + 9/4) + 9/4) dx`
`= int sqrt ((sqrt(13)/2)^2 - (x - 3/2)^2) dx`
`= [(x - 3/2)/2* sqrt (13/4 - (x - 3/2)^2) + 13/8 sin^-1 ((x - 3/2)/(sqrt 13/2))] +C` `....[int sqrt (a^2 - x^2) dx = x/2 sqrt (a^2 - x^2) + a^2/2 sin^-1 x/a + C]`
`= (2x - 3)/4 sqrt (1 + 3x - x^2) + 13/8 sin^-1 ((2x - 3)/sqrt13) + C`
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