Integralen av $$$\frac{\cos{\left(x \right)}}{54}$$$

Bestäm $$$\int \frac{\cos{\left(x \right)}}{54}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{54}$$$ och $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

$${\color{red}{\int{\frac{\cos{\left(x \right)}}{54} d x}}} = {\color{red}{\left(\frac{\int{\cos{\left(x \right)} d x}}{54}\right)}}$$

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

$$\frac{{\color{red}{\int{\cos{\left(x \right)} d x}}}}{54} = \frac{{\color{red}{\sin{\left(x \right)}}}}{54}$$

Alltså,

$$\int{\frac{\cos{\left(x \right)}}{54} d x} = \frac{\sin{\left(x \right)}}{54}$$

Lägg till integrationskonstanten:

$$\int{\frac{\cos{\left(x \right)}}{54} d x} = \frac{\sin{\left(x \right)}}{54}+C$$

$$$\int \frac{\cos{\left(x \right)}}{54}\, dx = \frac{\sin{\left(x \right)}}{54} + C$$$A