Question

$\int \frac{x}{\sqrt{4-x^4}}dx$

Answer 1

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