Integraal van $$$- \frac{\cos{\left(6 x \right)}}{6}$$$

Bepaal $$$\int \left(- \frac{\cos{\left(6 x \right)}}{6}\right)\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=- \frac{1}{6}$$$ en $$$f{\left(x \right)} = \cos{\left(6 x \right)}$$$:

$${\color{red}{\int{\left(- \frac{\cos{\left(6 x \right)}}{6}\right)d x}}} = {\color{red}{\left(- \frac{\int{\cos{\left(6 x \right)} d x}}{6}\right)}}$$

Zij $$$u=6 x$$$.

Dan $$$du=\left(6 x\right)^{\prime }dx = 6 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{6}$$$.

De integraal wordt

$$- \frac{{\color{red}{\int{\cos{\left(6 x \right)} d x}}}}{6} = - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{6}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{6}$$$ en $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$- \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{6} = - \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{6}\right)}}}{6}$$

De integraal van de cosinus is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$- \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{36} = - \frac{{\color{red}{\sin{\left(u \right)}}}}{36}$$

We herinneren eraan dat $$$u=6 x$$$:

$$- \frac{\sin{\left({\color{red}{u}} \right)}}{36} = - \frac{\sin{\left({\color{red}{\left(6 x\right)}} \right)}}{36}$$

Dus,

$$\int{\left(- \frac{\cos{\left(6 x \right)}}{6}\right)d x} = - \frac{\sin{\left(6 x \right)}}{36}$$

Voeg de integratieconstante toe:

$$\int{\left(- \frac{\cos{\left(6 x \right)}}{6}\right)d x} = - \frac{\sin{\left(6 x \right)}}{36}+C$$

$$$\int \left(- \frac{\cos{\left(6 x \right)}}{6}\right)\, dx = - \frac{\sin{\left(6 x \right)}}{36} + C$$$A