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