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Question
Integrate the function:
`sqrt(x^2 + 3x)`
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Solution
Let `I = int sqrt(x^2 + 3x)` dx
`= int sqrt((x^2 + 3x + 9/4) - 9/4)` dx
`= int sqrt((x + 3/2)^2 - (3/2)^2)` dx
`= ((x + 3/2))/2 sqrt((x + 3/2)^2 - 9/4) - 9/8 log abs ((x + 3/2) + sqrt((x + 3/2)^2 - 9/4)) + C` `....[∵ 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]`
`= (2x + 3)/4 sqrt (x^2 + 3x) - 9/8 log abs (x + 3/2 + sqrt(x^2 + 3x)) + C`
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