Integraal van $$$\frac{n x \sin{\left(c \right)}}{k}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \frac{n x \sin{\left(c \right)}}{k}\, dx$$$.

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

$${\color{red}{\int{\frac{n x \sin{\left(c \right)}}{k} d x}}} = {\color{red}{\frac{n \sin{\left(c \right)} \int{x d x}}{k}}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:

$$\frac{n \sin{\left(c \right)} {\color{red}{\int{x d x}}}}{k}=\frac{n \sin{\left(c \right)} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{k}=\frac{n \sin{\left(c \right)} {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{k}$$

Dus,

$$\int{\frac{n x \sin{\left(c \right)}}{k} d x} = \frac{n x^{2} \sin{\left(c \right)}}{2 k}$$

Voeg de integratieconstante toe:

$$\int{\frac{n x \sin{\left(c \right)}}{k} d x} = \frac{n x^{2} \sin{\left(c \right)}}{2 k}+C$$

$$$\int \frac{n x \sin{\left(c \right)}}{k}\, dx = \frac{n x^{2} \sin{\left(c \right)}}{2 k} + C$$$A