📝 题目
例 7 求 $\displaystyle{\int {\cos }^{3}x\mathrm{\;d}x,\int {\sin }^{3}x\mathrm{\;d}x}$ .
💡 答案与解析
解 $\displaystyle \int {\cos }^{3}x\mathrm{\;d}x = \int {\cos }^{2}x\mathrm{\;d}\left( {\sin x}\right)$
$$ = \int \left( {1 - {\sin }^{2}x}\right) \mathrm{d}\left( {\sin x}\right) $$
$$ = \sin x - \frac{1}{3}{\sin }^{3}x + C. $$
$$ \int {\sin }^{3}x\mathrm{\;d}x = - \int \left( {1 - {\cos }^{2}x}\right) \mathrm{d}\left( {\cos x}\right) $$
$$ = - \cos x + \frac{1}{3}{\cos }^{3}x + C. $$