解：

\[

\begin{flalign}

\int_{0}^{1}\frac{1}{(x + 1)(x^2 - 2x + 2)}dx 

&= \int_{0}^{1}\left(\frac{A}{x + 1}+\frac{Bx + C}{x^2 - 2x + 2}\right)dx \nonumber \\

&= \int_{0}^{1}\left(\frac{\frac{1}{5}}{x + 1}+\frac{-\frac{1}{5}x+\frac{3}{5}}{x^2 - 2x + 2}\right)dx \nonumber \\

&= \frac{1}{5}\ln|1 + x|\bigg|_{0}^{1} 

- \frac{1}{10}\ln|x^2 - 2x + 2|\bigg|_{0}^{1} 

+ \frac{2}{5}\arctan(x - 1)\bigg|_{0}^{1} \nonumber \\

&= \frac{3}{10}\ln2 + \frac{1}{10}\pi \nonumber

\end{flalign}

\]