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Question
Integrate the function `(x + 2)/sqrt(4x - x^2)`
Solution
Let `I = int (x + 2)/sqrt(4x - x^2) dx`
`= int (x + 2)/sqrt(- (x^2 - 4x + 4) + 4) dx`
`= int (x + 2)/sqrt(4 - (x - 2)^2) dx`
`= (x - 2 + 4)/sqrt(4 - (x - 2)^2) dx`
`= int (x - 2)/sqrt(4 - (x - 2)^2) dx + 4 int 1/sqrt(4 - (x - 2)^2) dx`
Let `I = I_1 + 4sin^-1 ((x - 2)/2) + C_1` .....(i)
Where `I_1 = int (x - 2)/ sqrt (4 - (x - 2)^2)dx`
Put 4 - (x - 2)2 = t
-2 (x - 2) dx = dt
`I_1 = 1/2 int dt/ sqrt ((2)^2 - t)`
`1/2 int dt/ sqrt (4 - t)`
`1/2 [(4 - t)^(-1/(2+1))/-(-1/2 + 1)] + C_2`
`= -sqrt ((4 - t)) + C_2`
`= - sqrt (4 - (x - 2)^2) + C_2`
`= - sqrt (4 - x^2 - 4 + 4x) + C_2`
`= - sqrt (4x - x^2) + C_2` ....(ii)
`I = - sqrt (4x - x^2) + 4sin^-1 ((x - 2)/2) + C` ... [from (i) and (ii)]
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