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Question
Integrate the following functions with respect to x :
`(x + 1)/((x + 2)(x + 3))`
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Solution
`(x + 1)/((x + 2)(x + 3)) = "A"/(x + 2) + "B"/(x + 3)`
Put x = – 3
– 3 + 1 = A(– 3 + 3) + B(– 3 + 2)
– 2 = A × 0 + B(– 1)
B = 2
Put x = – 2
– 2 + 1 = A(– 2 + 3) + B(– 2 + 2)
– 1 = A × 1 + B × 0
A = – 1
`(x + 1)/((x + 2)(x + 3)) = (- 1)/(x + 2) + 2/(x + 3)`
`int (x + 1)/((x + 2)(x + 3)) "d"x = int ((-1)/(x + 2) + 2/(x + 3)) "d"x`
= `- int ("d"x)/(x + 2) + 2int ("d"x)/(x + 3)`
= – log |x + 2| + 2 log |x + 3| + c
= 2 log |x + 3| – log |x + 2|+ c
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