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Question
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
Options
True
False
Solution
True
Explanation:
Let I =`(("x - 1"))/(("x + 1")("x - 2"))` dx
Let `(("x - 1"))/(("x + 1")("x - 2")) = "A"/"x + 1" + "B"/"x - 2"`
∴ x - 1 = A(x - 2) + B(x + 1) ....(i)
Putting x = –1 in (i), we get
- 1 - 1 = A(- 1 - 2)
∴ - 2 = - 3A
∴ A = `2/3`
Putting x = 2 in (i), we get
2 - 1 = B(2 + 1)
∴ 1 = 3B
∴ B = `1/3`
∴ I = `int ((2/3)/("x + 1") + (1/3)/("x - 2"))` dx
`= 2/3 int 1/"x + 1" "dx" + 1/3 int 1/"x - 2"` dx
`= 2/3 log |"x + 1"| + 1/3 log |"x - 2"|` + c
Comparing the above with
A log |x + 1| + B log |x - 2| + c, we get
∴ A = `2/3, "B" = 1/3`
∴ A + B = `2/3 + 1/3 = 1`
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