📝 题目
例 2 求 $\displaystyle{\int \frac{\mathrm{d}x}{{\sin }^{2}{2x}}}$ .
💡 答案与解析
解
$$ \int \frac{\mathrm{d}x}{{\sin }^{2}{2x}} = \frac{1}{4}\int \frac{\mathrm{d}x}{{\sin }^{2}x \cdot {\cos }^{2}x} $$
$$ = \frac{1}{4}\int \frac{{\sin }^{2}x + {\cos }^{2}x}{{\sin }^{2}x \cdot {\cos }^{2}x}\mathrm{\;d}x $$
$$ = \frac{1}{4}\left( {\int \frac{\mathrm{d}x}{{\cos }^{2}x}+\int \frac{\mathrm{d}x}{{\sin }^{2}x}}\right) $$
$$ = \frac{1}{4}\left( {\tan x - \cot x}\right) + C. $$
上式右端可以改写为
$$ \frac{{\sin }^{2}x - {\cos }^{2}x}{4\cos x\sin x} + C = - \frac{1}{2}\cot {2x} + C. $$
在下一节中, 我们将用更简单的办法求得这一结果.