📝 题目
3.1.5 求下列不定积分:
(1) $\displaystyle \int \ln \left( {1 + {x}^{2}}\right) \mathrm{d}x$ ; (2) $\displaystyle{\int {x}^{\alpha }\ln x\mathrm{\;d}x}$ ;
(3) $\displaystyle{\int \sqrt{x}\ln {}^{2}x\mathrm{\;d}x}$ ; (4) $\displaystyle{\int {x}^{2}{\mathrm{e}}^{-{2x}}\mathrm{\;d}x}$ ;
(5) $\displaystyle{\int x\cos {\beta x}\mathrm{\;d}x}$ ; (6) $\displaystyle{\int {x}^{2}\sin {2x}\mathrm{\;d}x}$ ;
(7) $\displaystyle{\int x\arctan x\mathrm{\;d}x}$ ; (8) $\displaystyle{\int \frac{\arcsin x}{{x}^{2}}\mathrm{\;d}x}$ ;
(9) $\displaystyle{\int \frac{x}{{\cos }^{2}x}\mathrm{\;d}x}$ ; (10) $\displaystyle \int \frac{{x}^{2}{\mathrm{e}}^{x}}{{\left( x + 2\right) }^{2}}\mathrm{\;d}x$ .
💡 答案与解析
3. 1.1 (1) $x - {\mathrm{e}}^{x} + \frac{1}{2}{\mathrm{e}}^{2x} + C$ ; (2) $- \arctan x - \frac{1}{x} + C$ ;
(3) $\frac{4}{7}{x}^{\frac{7}{4}} + C$ ; (4) $2\arcsin x + C$ ;
(5) $\tan x - x + C$ ;
(6) $\frac{1}{2\cos {2x} + 2}\left( {4\sin {2x} - {2x} - {2x}\cos {2x}}\right) + C$ ;
(7) $- 4\frac{\sin {4x}}{2 - 2\cos {4x}} + C$ ; (8) $- 8\frac{\sin {2x}}{2 - 2\cos {4x}} + C$ ；
(9) $\frac{1}{6}\sqrt{6}\arctan \frac{1}{2}x\sqrt{6} + C$ ;
(10) $\frac{1}{12}\sqrt{6}\ln \left( {x + \frac{1}{3}\sqrt{6}}\right) - \frac{1}{12}\sqrt{6}\ln \left( {x - \frac{1}{3}\sqrt{6}}\right) + C$ ;
(11) $\sqrt[3]{1 - {3x}}\left( {\frac{3}{4}x - \frac{1}{4}}\right) + C$ ;
(12) $\sqrt[3]{1 - {3x}}\left( {\frac{3}{7}{x}^{2} - \frac{1}{28}x - \frac{1}{28}}\right) + C$ .