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Question
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
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Solution
Let I = `int (2x + 1)/((x + 1)(x - 2)) "d"x`
Let `(2x + 1)/((x + 1)(x - 2)) = "A"/(x + 1) + "B"/(x - 2)`
∴ 2x + 1 = A(x – 2) + B(x + 1) ......(i)
Putting x = – 1 in (i), we get
2(– 1) + 1 = A(– 1 – 2) + B(0)
∴ – 1 = – 3A
∴ A = `1/3`
Putting x = 2 in (i), we get
2(2) + 1 = A(0) + B(2 + 1)
∴ 5 = 3B
∴ B = `5/3`
∴ `(2x + 1)/((x + 1)(x - 2)) = ((1/3))/(x + 1) + ((5/3))/(x - 2)`
∴ I = `int(((1/3))/(x + 1) + ((5/3))/(x - 2)) "d"x`
= `1/3 int 1/(x + 1) "d"x + 5/3 int 1/(x - 2) "d"x`
∴ I = `1/3 log|x + 1| + 5/3 log|x - 2| + "c"`
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