$$$- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1$$$の積分

$$$\int \left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)\, dx$$$ を求めよ。

項別に積分せよ:

$${\color{red}{\int{\left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{\sinh{\left(x \right)} d x} + \int{\cosh{\left(x \right)} d x}\right)}}$$

$$$c=1$$$ に対して定数則 $$$\int c\, dx = c x$$$ を適用する:

$$- \int{\sinh{\left(x \right)} d x} + \int{\cosh{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = - \int{\sinh{\left(x \right)} d x} + \int{\cosh{\left(x \right)} d x} + {\color{red}{x}}$$

双曲線正弦関数の不定積分は $$$\int{\sinh{\left(x \right)} d x} = \cosh{\left(x \right)}$$$:

$$x + \int{\cosh{\left(x \right)} d x} - {\color{red}{\int{\sinh{\left(x \right)} d x}}} = x + \int{\cosh{\left(x \right)} d x} - {\color{red}{\cosh{\left(x \right)}}}$$

双曲線余弦関数の積分は $$$\int{\cosh{\left(x \right)} d x} = \sinh{\left(x \right)}$$$ です:

$$x - \cosh{\left(x \right)} + {\color{red}{\int{\cosh{\left(x \right)} d x}}} = x - \cosh{\left(x \right)} + {\color{red}{\sinh{\left(x \right)}}}$$

したがって、

$$\int{\left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)d x} = x + \sinh{\left(x \right)} - \cosh{\left(x \right)}$$

簡単化せよ:

$$\int{\left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)d x} = x - e^{- x}$$

積分定数を加える:

$$\int{\left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)d x} = x - e^{- x}+C$$

$$$\int \left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)\, dx = \left(x - e^{- x}\right) + C$$$A