Integralen av $$$\frac{2 \sin{\left(x \right)}}{5}$$$

Bestäm $$$\int \frac{2 \sin{\left(x \right)}}{5}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{2}{5}$$$ och $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

$${\color{red}{\int{\frac{2 \sin{\left(x \right)}}{5} d x}}} = {\color{red}{\left(\frac{2 \int{\sin{\left(x \right)} d x}}{5}\right)}}$$

Integralen av sinus är $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

$$\frac{2 {\color{red}{\int{\sin{\left(x \right)} d x}}}}{5} = \frac{2 {\color{red}{\left(- \cos{\left(x \right)}\right)}}}{5}$$

Alltså,

$$\int{\frac{2 \sin{\left(x \right)}}{5} d x} = - \frac{2 \cos{\left(x \right)}}{5}$$

Lägg till integrationskonstanten:

$$\int{\frac{2 \sin{\left(x \right)}}{5} d x} = - \frac{2 \cos{\left(x \right)}}{5}+C$$

$$$\int \frac{2 \sin{\left(x \right)}}{5}\, dx = - \frac{2 \cos{\left(x \right)}}{5} + C$$$A