$$$- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1$$$ 的積分
相關計算器: 定積分與廣義積分計算器
您的輸入
求$$$\int \left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)\, dx$$$。
解答
逐項積分：
$${\color{red}{\int{\left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{\sinh{\left(x \right)} d x} + \int{\cosh{\left(x \right)} d x}\right)}}$$
配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：
$$- \int{\sinh{\left(x \right)} d x} + \int{\cosh{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = - \int{\sinh{\left(x \right)} d x} + \int{\cosh{\left(x \right)} d x} + {\color{red}{x}}$$
雙曲正弦函數的積分為 $$$\int{\sinh{\left(x \right)} d x} = \cosh{\left(x \right)}$$$：
$$x + \int{\cosh{\left(x \right)} d x} - {\color{red}{\int{\sinh{\left(x \right)} d x}}} = x + \int{\cosh{\left(x \right)} d x} - {\color{red}{\cosh{\left(x \right)}}}$$
雙曲餘弦的積分為 $$$\int{\cosh{\left(x \right)} d x} = \sinh{\left(x \right)}$$$：
$$x - \cosh{\left(x \right)} + {\color{red}{\int{\cosh{\left(x \right)} d x}}} = x - \cosh{\left(x \right)} + {\color{red}{\sinh{\left(x \right)}}}$$
因此，
$$\int{\left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)d x} = x + \sinh{\left(x \right)} - \cosh{\left(x \right)}$$
化簡：
$$\int{\left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)d x} = x - e^{- x}$$
加上積分常數：
$$\int{\left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)d x} = x - e^{- x}+C$$
答案
$$$\int \left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)\, dx = \left(x - e^{- x}\right) + C$$$A