$$$e^{x} - \sin{\left(x \right)}$$$の積分
入力内容
$$$\int \left(e^{x} - \sin{\left(x \right)}\right)\, dx$$$ を求めよ。
解答
項別に積分せよ:
$${\color{red}{\int{\left(e^{x} - \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{e^{x} d x} - \int{\sin{\left(x \right)} d x}\right)}}$$
正弦関数の不定積分は$$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$です:
$$\int{e^{x} d x} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = \int{e^{x} d x} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
指数関数の積分は $$$\int{e^{x} d x} = e^{x}$$$です:
$$\cos{\left(x \right)} + {\color{red}{\int{e^{x} d x}}} = \cos{\left(x \right)} + {\color{red}{e^{x}}}$$
したがって、
$$\int{\left(e^{x} - \sin{\left(x \right)}\right)d x} = e^{x} + \cos{\left(x \right)}$$
積分定数を加える:
$$\int{\left(e^{x} - \sin{\left(x \right)}\right)d x} = e^{x} + \cos{\left(x \right)}+C$$
解答
$$$\int \left(e^{x} - \sin{\left(x \right)}\right)\, dx = \left(e^{x} + \cos{\left(x \right)}\right) + C$$$A