Ta biến đổi:
$f(x)=\frac{x^{2000} }{(x^2+1)^{1000} }.\frac{x}{(x^2+1)^2} $
Khi đó:
$\int\limits f(x)dx=\int\limits \frac{x^{2000} }{(x^{2}+1)^{1000} } .\frac{x}{(x^2+1)^2} dx=\frac{1}{2}\int\limits (\frac{x^2}{x^2+1} )^{1000}.d( \frac{x^2}{x^2+1}) $
$=\frac{1}{2}\frac{1}{1001}(\frac{x^2}{x^2+1)})^{1001}+C=\frac{1}{2002}.(\frac{x^2}{x^2+1})^{1001}+C $