Question

$\int \frac{x}{\sqrt{x^4+a^4}}dx$

Answer 1

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