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Question
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Solution
Let I=`∫(x+1)/((x+2)(x+3))dx`
`(x+1)/((x+2)(x+3))=A/(x+2)+B/(x+3)`
`x+1=A(x+3)+B(x+2)` .........(i)
∴ Putting x = -2 in equation (i) we get
-1 = A
∴ A = -1
∴ Putting x = -3 in equation (i) we get
-2 = -B
∴ B = 2
∴(x+1)/((x+2)(x+3))=1/(x+2)+2/(x+3)
`∴ I=int[-1/(x+2)+2/(x+3)]dx`
`∴I=-log|x+2|+2log|x+3|+c`
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