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Question
Integrate the function `(5x + 3)/sqrt(x^2 + 4x + 10)`
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Solution
Let `I =int (5x +3)/ sqrt (x^2 + 4x + 10) dx`
Put 5x + 3
`= A [d/dx (x^2 + 4x + 10)] +B`
5x + 3
= A (2x + 4) + B ... (i)
Comparing the coefficient of x in (i), we get
5 = 2A
⇒ A = `5/2`
Comparing the constant terms in (i), we get
3 = 4A + B
⇒ B = -7
`I = int (5/2 (2x + 4) + (-7))/ sqrt (x^2 + 4x + 10) dx`
`= 5/2 int (2x + 4)/sqrt (x^2 + 4x + 10) dx - 7 int dx/ sqrt (x^2 + 4x + 10)`
`= 5/2 I_1 - 7I_2`
∴ `I = 5/2 I_1 - 7I_2` ....(ii)
Now, `I_1 = int (2x + 4)/ sqrt (x^2 + 4x + 10) dx`
Put x2 + 4x + 10 = t
⇒ (2x + 4) dx = dt
∴ `I_1 = int dt/sqrtt`
`= int t^(-1/2) dt = 2 sqrtt`
`= 2 sqrt (x^2 + 4x + 10) + C_1` ....(iii)
and `I_2 = int dx/ sqrt(x^2 + 4x + 10) `
`= int dx / sqrt((x + 2)^2 + (sqrt( 6))^2`
`= log |x + 2 + sqrt ((x + 2)^2 + (sqrt (6))^2)|`
`= log |x + 2 + sqrt (x^2 + 4x + 10)| + C_2` ....(iv)
Hence, from (ii), (iii) and (iv), we get
`I = 5 sqrt (x^2 + 4x + 10) - 7 log |x + 2 + sqrt (x^2 + 4x + 10)| + C`
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