Which theorem is applied in this integral?

Question

In my textbook, there is the following integral: $\int \frac{x^3}{1+x^2}dx$, the next line is:$\frac{1}{2}\int \frac{x^2}{1+x^2}dx^2$, my question is what exactly did the author do? Which theorem did he used? How did he increased the degree in the differential(how from $dx$, he got to $dx^2$)

Answer 1

It is just a (often misleading) way to perform the substitution $x=\sqrt{u}$, $dx=\frac{du}{2\sqrt{u}}$ $$ \int \frac{x^3}{1+x^2}\,dx = \frac{1}{2}\int \frac{u^{3/2}}{(1+u)\,u^{1/2}}\,du=\frac{1}{2}\int\frac{u}{1+u}\,du.$$ This clearly leads to: $$ \int \frac{x^3}{1+x^2}\,dx = C+\frac{x^2-\log(1+x^2)}{2}.$$