\(\int \frac{1}{x^{3}-1}dx=\int \frac{1}{(x-1)(x^{2}+x+1)}dx\)

\(=\frac{1}{3}\int (\frac{1}{x-1}-\frac{x+2}{x^{2}+x+1})dx\)

\(=\frac{1}{3}\log \vert x-1\vert -\frac{1}{6}\int \frac{2x+4}{x^{2}+x+1}dx\)

\(=\frac{1}{3}\log \vert x-1\vert -\frac{1}{6}\int \frac{(2x+1)+3}{x^{2}+x+1}dx\)

\(=\frac{1}{3}\log \vert x-1\vert -\frac{1}{6}\log \vert x^{2}+x+1\vert -\frac{1}{2}\int \frac{1}{x^{2}+x+1}dx\)

\(=\frac{1}{3}\log \vert x-1\vert -\frac{1}{6}\log \vert x^{2}+x+1\vert -\frac{1}{2}\int \frac{1}{(x+\frac{1}{2})^{2}+\frac{3}{4}}dx\)

　　　　　　　　※公式 \(\int \frac{1}{x^{2}+a^{2}}dx=\frac{1}{a}tan^{-1}\frac{x}{a}+C\) より

\(=\frac{1}{3}\log \vert x-1\vert -\frac{1}{6}\log \vert x^{2}+x+1\vert -\frac{1}{2}\int \frac{1}{(x+\frac{1}{2})^{2}+(\frac{\sqrt{3}}{2})^{2}}dx\)

\(=\frac{1}{3}\log \vert x-1\vert -\frac{1}{6}\log \vert x^{2}+x+1\vert -\frac{1}{2}\cdot \frac{1}{\frac{\sqrt{3}}{2}}\cdot \tan ^{-1}\frac{x+\frac{1}{2}}{\frac{\sqrt{3}}{2}}+C\)

\(=\frac{1}{3}\log \vert x-1\vert -\frac{1}{6}\log \vert x^{2}+x+1\vert -\frac{1}{\sqrt{3}}\cdot \tan ^{-1}\frac{2x+1}{\sqrt{3}}+C\) 　………（答）