📝 题目
例 2 求 $\displaystyle{\int x{\mathrm{e}}^{{x}^{2}}\mathrm{\;d}x,\int \frac{x}{1 + {x}^{4}}\mathrm{\;d}x,\int \frac{{x}^{2}}{{\cos }^{2}{x}^{3}}\mathrm{\;d}x}$ .
💡 答案与解析
解 $\displaystyle \int x{\mathrm{e}}^{{x}^{2}}\mathrm{\;d}x = \frac{1}{2}\int {\mathrm{e}}^{{x}^{2}}\mathrm{\;d}\left( {x}^{2}\right) = \frac{1}{2}{\mathrm{e}}^{{x}^{2}} + C$ .
$$ \int \frac{x}{1 + {x}^{4}}\mathrm{\;d}x = \frac{1}{2}\int \frac{\mathrm{d}\left( {x}^{2}\right) }{1 + {\left( {x}^{2}\right) }^{2}} = \frac{1}{2}\arctan {x}^{2} + C. $$
$$ \int \frac{{x}^{2}}{{\cos }^{2}{x}^{3}}\mathrm{\;d}x = \frac{1}{3}\int \frac{\mathrm{d}\left( {x}^{3}\right) }{{\cos }^{2}{x}^{3}} = \frac{1}{3}\tan {x}^{3} + C. $$
更一般地, 我们有公式:
$$ \int g\left( {x}^{k}\right) {x}^{k - 1}\mathrm{\;d}x = \frac{1}{k}\int g\left( {x}^{k}\right) \mathrm{d}\left( {x}^{k}\right) . $$