Question

`int (xdx)/((x - 1)(x - 2))` equals

Options

• `log |(x - 1)^2/(x - 2)| + c`

• `log |(x - 2)^2/(x - 1)| + c`

• `log |(x - 1)^2/(x - 2)|`

• `log|(x - 1)(x - 2) + c`

Answer 1

`log |(x - 2)^2/(x - 1)| + c`

Explanation:

`x/((x - 1)(x - 2)) = 1/(x - 1) + 2/(x - 2)`

`x = A(x - 2) + B(x - 1)`

Put `x = 1, 1 = A(1 - 2) = - A`    ∴ `A = - 1`

Put `x = 2, 2 = B(2 - 1) = B`     ∴ `B = 2`

∴ `x/((x - 1)(x - 2)) = 1/(x - 1) + 2/(x - 2)`

∴ `int (xdx)/((x - 1)(x - 2)) = - int 1/(x - 1) dx + 2 int 2/(x - 2) dx`

= `- log |x - 1| + 2 log |x - 2| + c`

= `log |(x + 2)^2/(x - 1)| + c`