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Question
Integrate the function `(x+2)/sqrt(x^2 + 2x + 3)`
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Solution
Let `I = int (x + 2)/ sqrt (x^2 + 2x + 3) dx`
`1/2 int (2x + 4)/ sqrt (x^2 + 2x + 3) dx`
`1/2 int (2x + 2 + 2)/sqrt (x^2 + 2x + 3) dx`
`1/2 int (2x + 2)/ sqrt(x^2 + 2x + 3) dx + dx/sqrt (x^2 + 2x + 3)`
Let I = I1 + I2 ....(i)
Where `I_1 = 1/2 int (2x + 2)/sqrt (x^2 + 2x + 3) dx`
Let x2 + 2x + 3 = t
⇒ (2x + 2) dx = dt
∴ `I_1 = 1/2 int dt/sqrtt `
`= 1/2 int t^(-1/2) dt = 1/2 xx 2t^(1/2)`
`= 1/2 xx 2 sqrt (x^2 + 2x + 3) + C_1`
`= sqrt (x^2 + 2x + 3) + C_1` .....(ii)
Also,
`I_2 = int dx/sqrt (x^2 + 2x + 3)`
`= int dx/sqrt (x^2 + 2x + 1 - 1 + 3)`
`= dx/ sqrt ((x + 1)^2 + sqrt( (2)^2))`
`log |(x + 1) + sqrt ((x + 1)^2 + 2)|`
`log |(x + 1) + sqrt (x^2 + 2x + 3)| + C_2` ....(iii)
Hence from (i), (ii) and (iii), we get
`I = sqrt (x^2 + 2x + 3) + log |(x + 1) + sqrt (x^2 + 2x + 3)| + C`
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