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Question
Evaluate the following:
`int sqrt(5 - 2x + x^2) "d"x`
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Solution
Let I = `int sqrt(5 - 2x + x^2) "d"x`
= `int sqrt(x^2 - 2x + 5) "d"x`
= `int sqrt(x^2 - 2x + 1 - 1 + 5) "d"x` ....(Making perfect square)
= `int sqrt((x - 1)^2 + 4) "d"x`
= `int sqrt((x - 1)^2 + (2)^2) "d"x`
= `(x - 1)/2 sqrt((x - 1)^2 + (2)^2) + 4/2 log|(x - 1) + sqrt((x + 1)^2 + (2)^2)| + "C"` .......`[because int sqrt(x^2 + "a"^2) "d"x = x/2 sqrt(x^2 + "a"^2) + "a"^2/2 {log|x + sqrt(x^2 + "a"^2)|} + "C"]`
= `(x - 1)/2 sqrt(x^2 + 1 - 2x + 4) + 2log |(x - 1) + sqrt(x - 1) + sqrt(x^2 + 1 - 2x + 4)| + "C"`
= `(x - 1)/2 sqrt(x^2 - 2x + 5) + 2log|(x - 1) + sqrt(x^2 - 2x + 5)| + "C"`
Hence, I = `(x - 1)/2 sqrt(x^2 - 2x + 5) + 2log|(x - 1) + sqrt(x^2 - 2x + 5)| + "C"`
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