How to find $\int \frac{2x}{x^4+1}dx$

Question

Can you give me a hint how to start solving this? $$\int \frac{2x}{x^4+1} dx$$

Answer 1

Hint: Use substitution $$ u = x^2. $$ so that $$ du = 2xdx $$

Answer 2

Find a way to apporach this from: $$(\arctan x)'=\frac{1}{x^2+1} $$

Answer 3

If we set $\color{red}{\bf u = x^2}$, then $\color{blue}{\bf du = 2x\,dx}$

$$\int \frac{2x}{x^4 + 1} dx = \int \frac{\color{blue}{\bf 2x}}{(\color{red}{\bf x^2})^2+1}\, \color{blue}{\bf dx} = \int \frac{1}{\color{red}{\bf u}^2 + 1} \,\color{blue}{\bf du} $$

Review your integrals to find the integral, given this form.

Hint: Can you recall the function $f(u)$ whose derivative is equal to $$f'(u) = \dfrac{1}{u^2 + 1}\;?$$