The integral (2x^3-1)/(x^4+x),dx is equal to : (Here C is a constant of integration)

Question

The integral $\displaystyle\int\frac{2x^3-1}{x^4+x}\,dx$ is equal to : (Here C is a constant of integration)

Accepted Answer

$$I=\int\frac{2x^3-1}{x^4+x}dx=\int\frac{2x-x^{-2}}{x^2+x^{-1}}dx$$

Let $x^2+x^{-1}=u\Rightarrow(2x-x^{-2})dx=du$:
$$I=\int\frac{du}u=\ln|u|+c=\ln\left|x^2+\frac1x\right|+c=\ln\left|\frac{x^3+1}x\right|+c$$

Solutions · 1