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Question
Integrate the following with respect to x.
`sqrt(2x^2 + 4x + 1)`
Solution
`int sqrt(2x^2 + 4x + 1) "d"x = int sqrt(2(x^2 +2x +1/2)) "d"x`
= `sqrt(2) int sqrt((x + 1)^2 - 1 + 1/2) "d"x`
= `sqrt(2) int sqrt((x + 1)^2 - 1/2) "d"x`
= `sqrt(2) int sqrt((x + 1)^2 - 1/2) "d"x`
= `sqrt(2) int sqrt((x + 1)^2 - (1/sqrt(2))^2) "d"x`
= `sqrt(2) [((x + 1))/2 sqrt((x + 1)^2 - 1/2) - (1/2)/2log|(x + 1) + sqrt((x + 1)^2 - 1/2)|] + "c"`
= `sqrt(2) [((x + 1))/2 sqrt((2x^2 + 4x + 1)/sqrt(2)) - 1/4 log|(x + 1)+sqrt(x^2 + 4x + 1)/sqrt(2)|] + "c"`
= `((x + 1)/2) sqrt(x^2 + x + 1) - sqrt(2)/4 log|sqrt(2)(x+ 1) + sqrt(x^2 + 4x + 1)| + "c"`
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