Funktion $$$\sin{\left(x \right)} + e$$$ integraali

Määritä $$$\int \left(\sin{\left(x \right)} + e\right)\, dx$$$.

Integroi termi kerrallaan:

$${\color{red}{\int{\left(\sin{\left(x \right)} + e\right)d x}}} = {\color{red}{\left(\int{e d x} + \int{\sin{\left(x \right)} d x}\right)}}$$

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

$$\int{\sin{\left(x \right)} d x} + {\color{red}{\int{e d x}}} = \int{\sin{\left(x \right)} d x} + {\color{red}{e x}}$$

Sinifunktion integraali on $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

$$e x + {\color{red}{\int{\sin{\left(x \right)} d x}}} = e x + {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Näin ollen,

$$\int{\left(\sin{\left(x \right)} + e\right)d x} = e x - \cos{\left(x \right)}$$

Lisää integrointivakio:

$$\int{\left(\sin{\left(x \right)} + e\right)d x} = e x - \cos{\left(x \right)}+C$$

$$$\int \left(\sin{\left(x \right)} + e\right)\, dx = \left(e x - \cos{\left(x \right)}\right) + C$$$A