Question

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

Answer 1

$$\begin{gathered} \int \left(\frac{x^4+x{}^2-1}{x^2+1}\right)dx \\ \Rightarrow \int \left(\frac{x^4+x^2}{x^2+1}\right)dx-\int \frac{1}{x^2+1}dx \\ \Rightarrow \int \frac{x^2\left(x^2+1\right)}{\left(x^2+1\right)}dx-\int \frac{1}{x^2+1}dx \\ \Rightarrow \int x^{2}dx-\int \frac{1}{x^2+1}dx \\ \Rightarrow \frac{x^3}{3}-{\tan }^{-1}\left(x\right)+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\because \int \frac{1}{x^{\mathit{2}}+a^{\mathit{2}}}dx\mathit{=}\frac{1}{a}{\tan }^{-1}\frac{x}{a}+C\right) \end{gathered}$$