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Question
Evaluate: `int sqrt(x^2 - 8x + 7)` dx
Solution
Let I = `int sqrt(x^2 − 8x + 7)` dx
`"I" = int sqrt(x^2 − 8x + 16 − 9)` dx
`"I" = int sqrt((x^2 − 8x + 16) − 9)` dx
`"I" = int sqrt((x − 4)^2 − (3)^2)` dx
∴ `int sqrt(x^2 − a^2) "dx" = x/2 sqrt(x^2 − a^2) − a^2/2 log |x + sqrt(x^2 − a^2)| + c`
`"I" = (x − 4)/2 sqrt((x - 4)^2 − (3)^2) − (3)^2/2log |(x − 4) + sqrt((x − 4)^2 − (3)^2)| + c`
`"I" = (x − 4)/2 sqrt(x^2 − 8x + 7) − 9/2 log |(x − 4) + sqrt(x^2 − 8x + 7)| + c`
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