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Question
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
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Solution
`I=∫(x+2)/sqrt(x^2+5x+6)dx `
Multiplying and dividing by 2, we get
`I=1/2∫(2x+4)/sqrt(x^2+5x+6)dx `
Adding and subtracting 1 to the numerator, we get:
`I=1/2∫(2x+4+1-1)/sqrt(x^2+5x+6)dx`
` I=1/2∫(2x+5)/sqrt(x2+5x+6)dx -1/2∫1/sqrt(x^2+5x+6)dx`
`"Let" I_1=1/2∫(2x+5)/sqrt(x^2+5x+6)dx `
Put x2+5x+6=t
Differentiating with respect to x, we get:
(2x+5)dx=dt
`I_1=intdt/sqrtt`
`I_1=2sqrtt+c`
`I_1=2sqrt(x^2+5x+6)+c`
`1/2 int "dt"/sqrt t =∫1/sqrt(x^2+5x+(5/2)^2-(5/2)^2+6)dx`
`1/2 int "dt"/sqrt t - 1/2 int "dx"/sqrt(x^2+5x+6 + (5/2)^2 - 25/4)dx`
`1/2 "t"^(1/2)/(1/2) int 1/sqrt((x+5/2)^2-(1/2)^2)dx`
`= 1/2 xx 2 xx "t"^(1/2) - 1/2 |"log" x + 5/2 + sqrt (x^2 + 5x + 6)| + "C"`
`= sqrt (x^2 + 5x + 6) - 1/2 "log" |sqrt (x^2 + 5x + 6)| + "C"`
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