$$$\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)}}$$$ 的積分
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您的輸入
求$$$\int \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)}}\, dx$$$。
解答
重寫被積函數:
$${\color{red}{\int{\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)}} d x}}} = {\color{red}{\int{\frac{1}{2} d x}}}$$
配合 $$$c=\frac{1}{2}$$$，應用常數法則 $$$\int c\, dx = c x$$$：
$${\color{red}{\int{\frac{1}{2} d x}}} = {\color{red}{\left(\frac{x}{2}\right)}}$$
因此，
$$\int{\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)}} d x} = \frac{x}{2}$$
加上積分常數：
$$\int{\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)}} d x} = \frac{x}{2}+C$$
答案
$$$\int \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(2 x \right)}}\, dx = \frac{x}{2} + C$$$A