$$$e^{x} - \sin{\left(x \right)}$$$ 的積分
您的輸入
求$$$\int \left(e^{x} - \sin{\left(x \right)}\right)\, dx$$$。
解答
逐項積分：
$${\color{red}{\int{\left(e^{x} - \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{e^{x} d x} - \int{\sin{\left(x \right)} d x}\right)}}$$
正弦函數的積分為 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$：
$$\int{e^{x} d x} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = \int{e^{x} d x} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
指數函數的積分為 $$$\int{e^{x} d x} = e^{x}$$$：
$$\cos{\left(x \right)} + {\color{red}{\int{e^{x} d x}}} = \cos{\left(x \right)} + {\color{red}{e^{x}}}$$
因此，
$$\int{\left(e^{x} - \sin{\left(x \right)}\right)d x} = e^{x} + \cos{\left(x \right)}$$
加上積分常數：
$$\int{\left(e^{x} - \sin{\left(x \right)}\right)d x} = e^{x} + \cos{\left(x \right)}+C$$
答案
$$$\int \left(e^{x} - \sin{\left(x \right)}\right)\, dx = \left(e^{x} + \cos{\left(x \right)}\right) + C$$$A