📝 题目
例 6 求 $\displaystyle{\int \frac{\mathrm{d}x}{{x}^{4} - 1}}$ .
💡 答案与解析
解 $\displaystyle \int \frac{\mathrm{d}x}{{x}^{4} - 1} = \int \frac{\mathrm{d}x}{\left( {{x}^{2} - 1}\right) \left( {{x}^{2} + 1}\right) }$
$$ = \frac{1}{2}\left( {\int \frac{\mathrm{d}x}{{x}^{2} - 1}-\int \frac{\mathrm{d}x}{{x}^{2} + 1}}\right) $$
$$ = \frac{1}{4}\ln \left| \frac{x - 1}{x + 1}\right| - \frac{1}{2}\arctan x + C. $$