📝 题目
例 7 求 $\displaystyle{\int \frac{{x}^{2} + x - 1}{{x}^{3} - 2{x}^{2} + x - 2}\mathrm{\;d}x}$ .
💡 答案与解析
解 $\displaystyle \int \frac{{x}^{2} + x - 1}{{x}^{3} - 2{x}^{2} + x - 2}\mathrm{\;d}x = \int \frac{\left( {{x}^{2} + 1}\right) + \left( {x - 2}\right) }{\left( {{x}^{2} + 1}\right) \left( {x - 2}\right) }\mathrm{d}x$
$$ = \int \frac{\mathrm{d}x}{x - 2} + \int \frac{\mathrm{d}x}{{x}^{2} + 1} $$
$$ = \ln \left| {x - 2}\right| + \arctan x + C. $$