Question

`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?

Options

• `tan^-1 ((x^2 + 1)/2) + "c"`

• tan^-1 (x²) + c

• tan^-1 (2x² - 1) + c

• `tan^-1 ((x^2 - 1)/x) + "c"`

Answer 1

`tan^-1 ((x^2 - 1)/x) + "c"`

Explanation:

Let I = `int (x^2 + 1)/(x^4 - x^2 + 1)`dx

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

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

put `x - 1/x` = t

`(1 + 1/x^2)`dx = dt

`therefore "I" = int "dt"/("t"^2 + 1) = tan^-1 ("t") + "c"`

`= tan^-1 (x - 1/x) + "c"`

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