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Question
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Solution
Let `(2x)/(x^2 + 3x + 2) = (2x)/((x + 1)(x + 2)`
`= A/(x + 1) + B/(x + 2)`
⇒ 2x = A(x + 2) = B (x + 1) ... (1)
Putting x = -1 in equation (1),
2(-1) = A (-1 + 2)
⇒ -2 = A
∴ A = -2
Putting x = -2 in equation (1),
2(-2) = B (-2 + 1)
⇒ B = 4
`therefore (2x)/(x^2 + 3x + 2) = (-2)/(x + 1) + 4/(x + 4)`
`therefore int (2x)/(x^2 + 3x + 2) dx`
`= -2 int dx /(x + 1) + 4 int dx /(x + 2)`
`= -2 log abs (x + 1) + 4 log abs (x + 2) + C`
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