<s>[INST] Evaluate $\int\frac{x^3}{(x^2+1)^3}dx$ $\int\frac{x^3}{(x^2+1)^3}dx$ what is my $z$? I know that $x=z^n$ and $n$ is the LCD of the exponents, but my lcd here is just one? so my $z$ is gonna be $z=x^2+1$? but I saw the notes of my classmate its $z^2=x^2+1$ can someone explain this thoroughly please
 [/INST] If you substitute $z^2=x^2+1$ , $2zdz=2xdx$ then $$\int\frac{x^3}{(x^2+1)^3}dx =\int\frac{z^2-1}{z^6}zdz = \int(\frac{1}{z^3}-\frac{1}{z^5})dz$$
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