Question

Q. Evaluate $\int \frac{x^4+x^2+1}{x^2-x+1}dx$.
Explain the steps.

Answer 1

Dear Student,

$$\begin{gathered} x^{2}canbewrittenas2x^2-x^2 \\ So,x^4+x^2+1=x^4+2x^2-x^2+1 \\ \mathit{U}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathbf{,}\mathbf{}{\left({\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\right)}^{\mathbf{2}}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}{\mathit{x}}^{\mathbf{4}}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{2}{\mathit{x}}^{\mathbf{2}} \\ {x}^4+x^2+1={\left(x^2+1\right)}^2-x^2 \end{gathered}$$

Regards