Integraal van $$$\frac{x}{a^{3}}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \frac{x}{a^{3}}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{a^{3}}$$$ en $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{\frac{x}{a^{3}} d x}}} = {\color{red}{\frac{\int{x d x}}{a^{3}}}}$$

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

$$\frac{{\color{red}{\int{x d x}}}}{a^{3}}=\frac{{\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{a^{3}}=\frac{{\color{red}{\left(\frac{x^{2}}{2}\right)}}}{a^{3}}$$

Dus,

$$\int{\frac{x}{a^{3}} d x} = \frac{x^{2}}{2 a^{3}}$$

Voeg de integratieconstante toe:

$$\int{\frac{x}{a^{3}} d x} = \frac{x^{2}}{2 a^{3}}+C$$

$$$\int \frac{x}{a^{3}}\, dx = \frac{x^{2}}{2 a^{3}} + C$$$A