Question

Integrate:

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

Answer 1

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

Let $x^2=t$

$\Rightarrow 2xdx=dt$

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

$=\frac{3}{4}\int \frac{dt}{2(\frac{1}{2}+t^2)}$

$=\frac{3}{4}\int \frac{dt}{(\frac{1}{\sqrt{2}}{)}^2+t^2}$

$=\frac{3}{4}\left[\sqrt{2}ta{n}^{-1}\sqrt{2}t\right]+C$

$=\frac{3}{2\sqrt{2}}ta{n}^{-1}\sqrt{2}t+C$