Question

Solve

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

Answer 1

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

$\Rightarrow \int \frac{x^2+1}{(x^2-x+1)(x^2+x+1)}dx$

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

$\Rightarrow \frac{1}{2}\int \frac{dx}{x^2-\frac{2x}{2}+\frac{1}{4}+{\left(\frac{\sqrt{3}}{2}\right)}^2}+\frac{1}{2}\int \frac{dx}{x^2+\frac{2x}{2}+\frac{1}{4}+{\left(\frac{\sqrt{3}}{2}\right)}^2}$

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

$\Rightarrow \frac{1}{2}\times \frac{2}{\sqrt{3}}\times {tan}^{-1}\left(\frac{x-1/2}{\sqrt{3}/2}\right)+\frac{1}{2}\times \frac{2}{\sqrt{3}}\times {tan}^{-1}\left(\frac{x+1/2}{\sqrt{3}}\right)+c$

$\Rightarrow \frac{1}{\sqrt{3}}{tan}^{-1}\left(\frac{2x-1}{\sqrt{3}}\right)+\frac{1}{\sqrt{3}}{tan}^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+c$