Question

Solve $\int \frac{x^3+x+1}{x^4+x^2+1}dx$

Answer 1

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

$\int \left[\frac{2x+1}{2\left(x^2+x+1\right)}+\frac{1}{2\left(x^2-x+1\right)}\right]dx$

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

$=\frac{1}{2}\int \frac{2x+1}{x^2+x+1}dx+\frac{1}{2}\int \frac{1}{{\left(x-\frac{1}{2}\right)}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}dx$

$=\frac{1}{2}\log \left|x^2+x+1\right|+\frac{1}{2}\times \frac{1}{\frac{\sqrt{3}}{2}}{\tan }^{-1}\frac{x-\frac{1}{2}}{\frac{\sqrt{3}}{2}}$

$=\frac{1}{2}\log \left|x^2+x+1\right|+\frac{1}{\sqrt{3}}{\tan }^{-1}\frac{2x-1}{\sqrt{3}}+c$