$$$\sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}$$$ 的积分
相关计算器: 定积分与广义积分计算器
您的输入
求$$$\int \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}\, dx$$$。
解答
设$$$u=\sin{\left(x \right)}$$$。
则$$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (步骤见»)，并有$$$\cos{\left(x \right)} dx = du$$$。
因此，
$${\color{red}{\int{\sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} d x}}} = {\color{red}{\int{\sin{\left(u \right)} d u}}}$$
正弦函数的积分为 $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:
$${\color{red}{\int{\sin{\left(u \right)} d u}}} = {\color{red}{\left(- \cos{\left(u \right)}\right)}}$$
回忆一下 $$$u=\sin{\left(x \right)}$$$:
$$- \cos{\left({\color{red}{u}} \right)} = - \cos{\left({\color{red}{\sin{\left(x \right)}}} \right)}$$
因此，
$$\int{\sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} d x} = - \cos{\left(\sin{\left(x \right)} \right)}$$
加上积分常数：
$$\int{\sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} d x} = - \cos{\left(\sin{\left(x \right)} \right)}+C$$
答案
$$$\int \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}\, dx = - \cos{\left(\sin{\left(x \right)} \right)} + C$$$A