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