Funktion $$$- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1$$$ integraali

Määritä $$$\int \left(- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1\right)\, dx$$$.

Integroi termi kerrallaan:

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

Sovella vakiosääntöä $$$\int c\, dx = c x$$$ käyttäen $$$c=1$$$:

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

Hyperbolisen sinin integraali on $$$\int{\sinh{\left(x \right)} d x} = \cosh{\left(x \right)}$$$:

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

Hyperbolisen kosinin integraali on $$$\int{\cosh{\left(x \right)} d x} = \sinh{\left(x \right)}$$$:

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

Näin ollen,

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

Sievennä:

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

Lisää integrointivakio:

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

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