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Question
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
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Solution
`(5x)/((x + 1)(x^2 - 4))`
`= 5/((x + 1)(x + 2)(x - 2))`
`(5x)/((x + 1)(x^2 - 4)) => A/(x + 1) + B/(x + 2) + C/(x - 2)`
⇒ 5x = A(x2 - 4) + B (x + 1)(x - 2) + C(x + 1)(x + 2)
Put x = -1
-5 = -3A + 0 = 0
⇒ A `= 5/3`
Put x = -2
-10 = 0 + B(-1)(-4) + 0
⇒ B `= (-5)/2`
Put x = 2
10 = 0 + 0 + 12C
⇒ C `= 5/6`
`therefore int (5x)/((x + 1)(x^2 - 4)`
`= 5/3 int 1/(x + 1) dx - 5/2 int 1/(x + 1) dx + 5/6 int 1/(x - 2) dx`
`= 5/3 log abs (x + 1) - 5/2 log abs (x + 1) + 5/6 log abs (x - 2) + C`
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