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Question
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
Solution
`int x/((x + 2)(x + 3)) dx = bb(int (-2)/(x + 2))dx + int 3/(x + 3) dx`
Explanation:
Let `x/((x + 2)(x + 3)) = A/(x + 2) + B/(x + 3)`
⇒ x = A(x + 3) + B(x + 2)
⇒ x = (A + B)x + (3A + 2B)
On equating coefficients of like terms, we get
A + B = 1 .......(1)
⇒ B = 1 – A
⇒ B = 1 – (– 2) = 3
⇒ B = 3
And 3A + 2B = 0 ......(2)
⇒ 3A + 2(1 – A) = 0
⇒ 3A + 2 – 2A = 0
⇒ A + 2 = 0
⇒ A = – 2
∴ `int x/((x + 2)(x + 3)) dx = int (-2)/(x + 2) dx + int 3/(x + 3) dx`
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