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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"`
By partial fraction
`(2"x" + 1)/(("x" + 1)("x"-2)) = "A"/("x" + 1) + "B"/("x" - 2)`
∴ 2x + 1= A (x - 2) + B(x + 1)
Put x = -1
2 (-1) + 1 = A(x - 2) + B (0)
⇒ -1 = -3 A
∴ A = `1/3`
Put x = 2
2(2) + 1 = A(O) + B(2+1)
⇒ 5 = 3 B
∴ B = `5/3`
∴ 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"`
`= 1/3 "log" |"x" + 1| + 5/3 "log" |"x" - 2| + c`
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