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