Question

Evaluate $\int \frac{3x}{1+2x^4}dx$

Answer 1

$\int \frac{3x}{1+2x^4}dx$

$\int \frac{3x}{{\left(\sqrt{2}{x}^2\right)}^2+1}$

Putting $\sqrt{2}{x}^2=t$

$2\sqrt{2}xdx=dt$

$xdx=\frac{dt}{2\sqrt{2}}$

$\int \frac{3}{2\sqrt{2}}\times \frac{dt}{t^2+1}$

$\frac{3}{2\sqrt{2}}\int \frac{dt}{t^2+1}$

$\frac{3}{2\sqrt{2}}{\tan }^{-1}\left(t\right)+c$

$=\frac{3}{2\sqrt{2}}{\tan }^{-1}\left(\sqrt{2}{x}^2\right)+c$