How do I evaluate the following integrals?

Question

$$I=\int \frac{x^{2000}}{x^{2668}+1}\mathrm dx,\qquad J=\int \frac{x^{666}}{x^{2668}+1}\mathrm dx$$ I thought about doing $I+J$ and $I-J$ but I do not have any idea what to do after. Can you help me how should I evaluate this type of integrals?

Answer 1

Hint: $$\begin{align} & I-J=\int{\frac{{{x}^{2000}}-{{x}^{666}}}{{{x}^{2668}}+1}dx}=\int{\frac{1}{x}\frac{{{x}^{2001}}-{{x}^{667}}}{{{x}^{2668}}+1}dx} \\ & =\int{\frac{1}{x}\frac{{{\left( {{x}^{667}} \right)}^{3}}-{{x}^{667}}}{{{\left( {{x}^{667}} \right)}^{4}}+1}dx},\ \ Now\ use\ u={{x}^{667}}, \\ & =\frac{1}{667}\int{\frac{{{u}^{2}}-1}{{{u}^{4}}+1}du}=\frac{1}{667}\frac{\ln \left( -{{u}^{2}}+\sqrt{2}u-1 \right)-\ln \left( {{u}^{2}}+\sqrt{2}u+1 \right)}{2\sqrt{2}}. \\ \end{align}$$

Answer 2

By variable substitution. In both of them, try $$ x=u^{a} $$ for some $a$. If you need I can provide the $a$ that I used.