Question

Evaluate the following integrals : $\int \frac{x^2}{x^4+x^2+1}dx$

Answer 1

We have,

$\begin{array}{c}I=\int \frac{x^2}{x^4+x^2+1}dx\\ =\int \frac{dx}{x^2+1+\frac{1}{x^2}}\\ =\int \frac{dx}{{\left(x^2+\frac{1}{x}\right)}^2-1}\\ =\frac{1}{2}\log \left(\frac{x+\frac{1}{x}-1}{x+\frac{1}{x}+1}\right)\\ =\frac{1}{2}\log \left(\frac{x^2-x+1}{x^2+x+1}\right)+C\end{array}$

Hence, this is the answer.