Integraal van $$$\left(y - 4\right)^{2}$$$

Bepaal $$$\int \left(y - 4\right)^{2}\, dy$$$.

Zij $$$u=y - 4$$$.

Dan $$$du=\left(y - 4\right)^{\prime }dy = 1 dy$$$ (de stappen zijn te zien »), en dan geldt dat $$$dy = du$$$.

Dus,

$${\color{red}{\int{\left(y - 4\right)^{2} d y}}} = {\color{red}{\int{u^{2} d u}}}$$

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 $$$u=y - 4$$$:

$$\frac{{\color{red}{u}}^{3}}{3} = \frac{{\color{red}{\left(y - 4\right)}}^{3}}{3}$$

Dus,

$$\int{\left(y - 4\right)^{2} d y} = \frac{\left(y - 4\right)^{3}}{3}$$

Voeg de integratieconstante toe:

$$\int{\left(y - 4\right)^{2} d y} = \frac{\left(y - 4\right)^{3}}{3}+C$$

$$$\int \left(y - 4\right)^{2}\, dy = \frac{\left(y - 4\right)^{3}}{3} + C$$$A