integration - solving $intfrac{2x^3}{x^2+1} , dx$

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

$$u=x^2$$

$$du=2x \, dx$$

$$\int \frac{u}{u+1} \, du$$ $$=\int \frac{u+1-1}{u+1}du$$ $$=\int 1-\frac{1}{u+1} \, du$$

how should I continue? is there an algotherm for integrating rational functions?

Answer

Notice, here is another simple method, $$\int \frac{2x^3}{x^2+1}\ dx$$

$$\int \frac{2x^3+2x-2x}{x^2+1}\ dx$$
$$=\int \frac{2x(x^2+1)-2x}{x^2+1}\ dx$$
$$=\int 2x\ dx-\int \frac{2x}{x^2+1}\ dx$$
$$=2\int x\ dx-\int \frac{d(x^2+1)}{x^2+1}$$
$$=\color{red}{x^2-\ln(x^2+1)+C}$$