Integraal van $$$\left(\frac{x}{2} - 3\right)^{5}$$$

Bepaal $$$\int \left(\frac{x}{2} - 3\right)^{5}\, dx$$$.

Zij $$$u=\frac{x}{2} - 3$$$.

Dan $$$du=\left(\frac{x}{2} - 3\right)^{\prime }dx = \frac{dx}{2}$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = 2 du$$$.

Dus,

$${\color{red}{\int{\left(\frac{x}{2} - 3\right)^{5} d x}}} = {\color{red}{\int{2 u^{5} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=2$$$ en $$$f{\left(u \right)} = u^{5}$$$:

$${\color{red}{\int{2 u^{5} d u}}} = {\color{red}{\left(2 \int{u^{5} d u}\right)}}$$

Pas de machtsregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=5$$$:

$$2 {\color{red}{\int{u^{5} d u}}}=2 {\color{red}{\frac{u^{1 + 5}}{1 + 5}}}=2 {\color{red}{\left(\frac{u^{6}}{6}\right)}}$$

We herinneren eraan dat $$$u=\frac{x}{2} - 3$$$:

$$\frac{{\color{red}{u}}^{6}}{3} = \frac{{\color{red}{\left(\frac{x}{2} - 3\right)}}^{6}}{3}$$

Dus,

$$\int{\left(\frac{x}{2} - 3\right)^{5} d x} = \frac{\left(\frac{x}{2} - 3\right)^{6}}{3}$$

Vereenvoudig:

$$\int{\left(\frac{x}{2} - 3\right)^{5} d x} = \frac{\left(x - 6\right)^{6}}{192}$$

Voeg de integratieconstante toe:

$$\int{\left(\frac{x}{2} - 3\right)^{5} d x} = \frac{\left(x - 6\right)^{6}}{192}+C$$

$$$\int \left(\frac{x}{2} - 3\right)^{5}\, dx = \frac{\left(x - 6\right)^{6}}{192} + C$$$A