The integral of x^2-xx^3-x^2+x-1 w.r.t.

Question

The integral of $\dfrac{x^2-x}{x^3-x^2+x-1}$ w.r.t. $x$ is:

Accepted Answer

$I = \int \dfrac{x^2-x}{x^3-x^2+x-1}\,dx = \int \dfrac{x(x-1)}{x^2(x-1)+(x-1)}\,dx = \int \dfrac{x\,dx}{x^2+1}$

$= \dfrac12\int \dfrac{2x\,dx}{x^2+1}$

Let $x^2+1=t \Rightarrow 2x\,dx=dt$

$I = \dfrac12\int\dfrac{dt}{t} = \dfrac12\log t+c = \dfrac12\log(x^2+1)+c$

Solutions · 1