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Question
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
Options
`1/2 log [(2x - 1) + sqrt(x^2 - x - 6)] + "c"`
tan−1 (2x − 1) + c
`log [(x - 1/2) + sqrt(x^2 - x - 6)] + "c"`
`log [(x - 1/2) + sqrt(x^2 + x + 6)] + "c"`
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Solution
`int 1/sqrt((x - 3)(x + 2))` dx = `bbunderline(log [(x - 1/2) + sqrt(x^2 - x - 6)] + "c")`.
Explanation:
`int 1/sqrt((x - 3)(x + 2))` dx = `int 1/sqrt(x^2 - x - 6)` dx
= `int 1/sqrt((x - 1/2)^2 - (5/2)^2)` dx
= `log |(x - 1/2) + sqrt((x - 1/2)^2 - ( 5/2)^2)| + c`
= `log |(x - 1/2) + sqrt(x^2 - x - 6)| + c`
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