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Question
Integrate the function `1/sqrt(8+3x - x^2)`
Solution
Let `I = int 1/sqrt(8 + 3x - x^2) dx`
`= int dx/sqrt(8 - (x^2 - 3x))`
`= int dx/sqrt(8 - (x^2 - 2 * 3/2 x + 9/4) + 9/4)`
`= int dx/sqrt(41/4 - (x - 3/2)^2)`
`= int dx/sqrt((sqrt41/2)^2 - (x - 3/2)^2)` `...[∵ int dx/sqrt (a^2 - x^2) = sin^-1 x/a + C]`
`= sin^-1 ((x - 3/2)/(sqrt41/2)) + C`
`= sin^-1 ((2x - 3)/sqrt41) + C`
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