Question

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

Answer 1

$$\begin{gathered} \int \frac{xdx}{x^4+2x^2+3} \\ \mathrm{let}{x}^2=t \\ \Rightarrow 2xdx=dt \\ \Rightarrow xdx=\frac{dt}{2} \\ \mathrm{Now},\int \frac{xdx}{x^4+2x^2+3} \\ =\frac{1}{2}\int \frac{dt}{t^2+2t+3} \\ =\frac{1}{2}\int \frac{dt}{t^2+2t+1+2} \\ =\frac{1}{2}\int \frac{dt}{{\left(t+1\right)}^2+{\left(\sqrt{2}\right)}^2} \\ =\frac{1}{2}\times \frac{1}{\sqrt{2}}{\tan }^{-1}\left(\frac{t+1}{\sqrt{2}}\right)+C\left[\because \int \frac{dx}{x^2+a^2}=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+C\right] \\ =\frac{1}{2\sqrt{2}}{\tan }^{-1}\left(\frac{x^2+1}{\sqrt{2}}\right)+C \end{gathered}$$