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Question
Integrate the function:
`sqrt(x^2 + 4x +1)`
Solution
Let `I = int sqrt(x^2 + 4x + 1)` dx
`= int sqrt((x^2 + 4x + 4) - 3)` dx
`= int sqrt((x + 2)^2 - (sqrt3)^2)` dx
`= ((x + 2))/2 sqrt((x + 2)^2 - 3) - 3/2 log abs ((x + 2) + sqrt ((x + 2)^2 + 3)) + C` `...[∵ sqrt (x^2 - a^2) dx = x/2 sqrt (x^2 - a^2) - a^2/2 log |x + sqrt (x^2 - a^2)| + C]`
`= ((x + 2))/2 sqrt(x^2 + 4x + 1) - 3/2 log abs((x + 2) + sqrt(x^2 + 4x + 1)) + C`
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