$$$\frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}}$$$ 的积分
相关计算器: 定积分与广义积分计算器
您的输入
求$$$\int \frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx$$$。
解答
改写被积函数:
$${\color{red}{\int{\frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{12 \cos{\left(x \right)} d x}}}$$
对 $$$c=12$$$ 和 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$${\color{red}{\int{12 \cos{\left(x \right)} d x}}} = {\color{red}{\left(12 \int{\cos{\left(x \right)} d x}\right)}}$$
余弦函数的积分为 $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$：
$$12 {\color{red}{\int{\cos{\left(x \right)} d x}}} = 12 {\color{red}{\sin{\left(x \right)}}}$$
因此，
$$\int{\frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 12 \sin{\left(x \right)}$$
加上积分常数：
$$\int{\frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 12 \sin{\left(x \right)}+C$$
答案
$$$\int \frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx = 12 \sin{\left(x \right)} + C$$$A