Question

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

Answer 1

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

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

$2\sqrt{2}xdx=dt$

Therefore,

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

$I=\frac{3}{2\sqrt{2}}{\tan }^{-1}t+C$ ....... $\because \int \frac{dx}{x^2+a^2}={\tan }^{-1}\frac{x}{a}+C$

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