Integraal van $$$- \sinh{\left(x \right)} + \cosh{\left(x \right)} + 1$$$

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

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=1$$$:

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

De integraal van de hyperbolische sinus is $$$\int{\sinh{\left(x \right)} d x} = \cosh{\left(x \right)}$$$:

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

De integraal van de cosinus hyperbolicus is $$$\int{\cosh{\left(x \right)} d x} = \sinh{\left(x \right)}$$$:

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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