$$$\sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}$$$ 的積分
相關計算器: 定積分與廣義積分計算器
您的輸入
求$$$\int \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}\, dx$$$。
解答
令 $$$u=\sin{\left(x \right)}$$$。
則 $$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\cos{\left(x \right)} dx = du$$$。
因此，
$${\color{red}{\int{\sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} d x}}} = {\color{red}{\int{\sin{\left(u \right)} d u}}}$$
正弦函數的積分為 $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$：
$${\color{red}{\int{\sin{\left(u \right)} d u}}} = {\color{red}{\left(- \cos{\left(u \right)}\right)}}$$
回顧一下 $$$u=\sin{\left(x \right)}$$$：
$$- \cos{\left({\color{red}{u}} \right)} = - \cos{\left({\color{red}{\sin{\left(x \right)}}} \right)}$$
因此，
$$\int{\sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} d x} = - \cos{\left(\sin{\left(x \right)} \right)}$$
加上積分常數：
$$\int{\sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} d x} = - \cos{\left(\sin{\left(x \right)} \right)}+C$$
答案
$$$\int \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}\, dx = - \cos{\left(\sin{\left(x \right)} \right)} + C$$$A