$$\int\frac{2x^3}{x^2+1} \, dx$$
$$u=x^2$$
$$du=2x \, dx$$
$$\int \frac{u}{u+1} \, du$$$$=\int \frac{u+1-1}{u+1}du$$$$=\int 1-\frac{1}{u+1} \, du$$
how should I continue? is there an algotherm for integrating rational functions?
Answer
Notice, here is another simple method, $$\int \frac{2x^3}{x^2+1}\ dx$$
$$\int \frac{2x^3+2x-2x}{x^2+1}\ dx$$
$$=\int \frac{2x(x^2+1)-2x}{x^2+1}\ dx$$
$$=\int 2x\ dx-\int \frac{2x}{x^2+1}\ dx$$
$$=2\int x\ dx-\int \frac{d(x^2+1)}{x^2+1}$$
$$=\color{red}{x^2-\ln(x^2+1)+C}$$
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