Integraal van $$$\left(a - x\right)^{2}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(a - x\right)^{2}\, dx$$$.

Zij $$$u=a - x$$$.

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

De integraal wordt

$${\color{red}{\int{\left(a - x\right)^{2} d x}}} = {\color{red}{\int{\left(- u^{2}\right)d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=-1$$$ en $$$f{\left(u \right)} = u^{2}$$$:

$${\color{red}{\int{\left(- u^{2}\right)d u}}} = {\color{red}{\left(- \int{u^{2} d u}\right)}}$$

Pas de machtsregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:

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

We herinneren eraan dat $$$u=a - x$$$:

$$- \frac{{\color{red}{u}}^{3}}{3} = - \frac{{\color{red}{\left(a - x\right)}}^{3}}{3}$$

Dus,

$$\int{\left(a - x\right)^{2} d x} = - \frac{\left(a - x\right)^{3}}{3}$$

Vereenvoudig:

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

Voeg de integratieconstante toe:

$$\int{\left(a - x\right)^{2} d x} = \frac{\left(- a + x\right)^{3}}{3}+C$$

$$$\int \left(a - x\right)^{2}\, dx = \frac{\left(- a + x\right)^{3}}{3} + C$$$A