Question

Find: `int x^4/((x - 1)(x^2 + 1))dx`.

Answer 1

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

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

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

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

= `x^2/2 + x + 1/2 int (1/(x - 1) - (x + 1)/(x^2 + 1))dx`  ...{∵ Partial factorisation}

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

= `x^2/2 + x + 1/2 log(x - 1) - 1/4 log (x^2 + 1) - 1/2 tan^-1 x + C`.