📝 题目
3. 1.9 求下列不定积分的递推公式:
(1) $\displaystyle{\int {x}^{n}{\mathrm{e}}^{x}\mathrm{\;d}x}$ ; (2) $\displaystyle \int {x}^{n}{\left( \ln x\right) }^{m}\mathrm{\;d}x$ ;
(3) $\displaystyle{\int {\sin }^{n}x\mathrm{\;d}x}$ ; (4) $\displaystyle \int \frac{\mathrm{d}x}{{\sin }^{n}x}\left( {n \geq 2}\right)$ .
💡 答案与解析
3. 1.9 (1) ${I}_{n} = {x}^{n}{\mathrm{e}}^{x} - n{I}_{n - 1}$ ;
(2) ${I}_{n,m} = \frac{1}{n + 1}{x}^{n + 1}{\left( \ln x\right) }^{m} - \frac{m}{n + 1}{I}_{n,m - 1}$ ;
(3) ${I}_{n} = - \frac{{\sin }^{n - 1}x\cos x}{n} + \frac{n - 1}{n}{I}_{n - 2}$ ;
(4) ${I}_{n} = - \frac{\cos x}{\left( {n - 1}\right) {\sin }^{n - 1}x} + \frac{n - 2}{n - 1}{I}_{n - 2}$ .