10. Evaluate the definite integral $\int_{-1}^{+1} \frac{2 u^{332}+u^{998}+4 u^{1664} \sin u^{691}}{1+u^{666}} \mathrm{~d} u$.
Solution: The term $\frac{4 u^{1664} \sin u^{991}}{1+u^{666}}$ is odd in $u$, so its integral is 0 . Now make the substitution $u=v^{1 / 333} \Rightarrow \mathrm{d} u=\frac{1}{333} v^{-332 / 333} \mathrm{~d} v$ to find that $\int_{-1}^{+1} \frac{2 u^{332}+u^{998}}{1+u^{666}} \mathrm{~d} u=\frac{1}{333} \int_{-1}^{+1} \frac{2+v^{2}}{1+v^{2}} \mathrm{~d} v=$ $\frac{1}{333} \int_{-1}^{+1}\left(1+\frac{1}{1+v^{2}}\right) \mathrm{d} v=\frac{2}{333} \int_{0}^{1}\left(1+\frac{1}{1+v^{2}}\right) \mathrm{d} v=\frac{2}{333}\left(1+\int_{0}^{1} \frac{1}{1+v^{2}} \mathrm{~d} v\right)=\frac{2}{333}\left(1+\tan ^{-1} 1\right)=\frac{2}{333}\left(1+\frac{\pi}{4}\right)$.