📝 题目
例 18 试利用公式
$$ \int \left( {f\left( x\right) + {f}^{\prime }\left( x\right) }\right) {\mathrm{e}}^{x}\mathrm{\;d}x = \int {\left( {\mathrm{e}}^{x}f\left( x\right) \right) }^{\prime }\mathrm{d}x = {\mathrm{e}}^{x}f\left( x\right) + C $$
求下列不定积分:
(1) $\displaystyle \int \frac{x{\mathrm{e}}^{x}}{{\left( 1 + x\right) }^{2}}\mathrm{\;d}x$ ; (2) $\displaystyle{\int \frac{1 + \sin x}{1 + \cos x}{\mathrm{e}}^{x}\mathrm{\;d}x}$ .
💡 答案与解析
解 (1) 原式 $= \int \frac{x + 1 - 1}{{\left( 1 + x\right) }^{2}}{\mathrm{e}}^{x}\mathrm{\;d}x = \int \left\lbrack {\frac{1}{1 + x} + {\left( \frac{1}{1 + x}\right) }^{\prime }}\right\rbrack {\mathrm{e}}^{x}\mathrm{\;d}x$
$$ = \frac{{\mathrm{e}}^{x}}{1 + x} + C. $$
(2)原式 $= \int \frac{2\sin \frac{x}{2}\cos \frac{x}{2} + 1}{2{\cos }^{2}\frac{x}{2}}{\mathrm{e}}^{x}\mathrm{\;d}x = \int \left\lbrack {\tan \frac{x}{2} + {\left( \tan \frac{x}{2}\right) }^{\prime }}\right\rbrack {\mathrm{e}}^{x}\mathrm{\;d}x$
$$ = {\mathrm{e}}^{x}\tan \frac{x}{2} + C. $$