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Question
Integrate the function `1/sqrt((x - a)(x - b))`
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Solution
Let `I = int dx/sqrt((x - a) (x - b))`
`= int dx/sqrt(x^2 - (a + b)x + ab)`
`= int dx/{[x^2 - 2 ((a + b)/2) x + ((a + b)/2)^2] + ab - ((a + b)/2)^2`
`= int dx/sqrt((x - (a + b)/2)^2 - ((a - b)/2)^2)`
`= log [(x - (a + b)/2) + sqrt((x - (a + b)/2)^2 - ((a - b)/2)^2)] + C` `...[∵ int dx/ sqrt (x^2 - a^2) = log |x + sqrt (x^2 - a^2)| + C]`
`= log [x - (a + b)/2 + sqrt((x - a)(x - b))] + C`
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