Integraal van $$$\sin{\left(10 x \right)}$$$

Bepaal $$$\int \sin{\left(10 x \right)}\, dx$$$.

Zij $$$u=10 x$$$.

Dan $$$du=\left(10 x\right)^{\prime }dx = 10 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{10}$$$.

Dus,

$${\color{red}{\int{\sin{\left(10 x \right)} d x}}} = {\color{red}{\int{\frac{\sin{\left(u \right)}}{10} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{10}$$$ en $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\sin{\left(u \right)}}{10} d u}}} = {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{10}\right)}}$$

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

$$\frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{10} = \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{10}$$

We herinneren eraan dat $$$u=10 x$$$:

$$- \frac{\cos{\left({\color{red}{u}} \right)}}{10} = - \frac{\cos{\left({\color{red}{\left(10 x\right)}} \right)}}{10}$$

Dus,

$$\int{\sin{\left(10 x \right)} d x} = - \frac{\cos{\left(10 x \right)}}{10}$$

Voeg de integratieconstante toe:

$$\int{\sin{\left(10 x \right)} d x} = - \frac{\cos{\left(10 x \right)}}{10}+C$$

$$$\int \sin{\left(10 x \right)}\, dx = - \frac{\cos{\left(10 x \right)}}{10} + C$$$A