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

Answer 1

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`