Integraal van $$$x^{2} \left(x^{3} - 1\right)^{10}$$$

Bepaal $$$\int x^{2} \left(x^{3} - 1\right)^{10}\, dx$$$.

Zij $$$u=x^{3} - 1$$$.

Dan $$$du=\left(x^{3} - 1\right)^{\prime }dx = 3 x^{2} dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$x^{2} dx = \frac{du}{3}$$$.

De integraal kan worden herschreven als

$${\color{red}{\int{x^{2} \left(x^{3} - 1\right)^{10} d x}}} = {\color{red}{\int{\frac{u^{10}}{3} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{3}$$$ en $$$f{\left(u \right)} = u^{10}$$$:

$${\color{red}{\int{\frac{u^{10}}{3} d u}}} = {\color{red}{\left(\frac{\int{u^{10} d u}}{3}\right)}}$$

Pas de machtsregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=10$$$:

$$\frac{{\color{red}{\int{u^{10} d u}}}}{3}=\frac{{\color{red}{\frac{u^{1 + 10}}{1 + 10}}}}{3}=\frac{{\color{red}{\left(\frac{u^{11}}{11}\right)}}}{3}$$

We herinneren eraan dat $$$u=x^{3} - 1$$$:

$$\frac{{\color{red}{u}}^{11}}{33} = \frac{{\color{red}{\left(x^{3} - 1\right)}}^{11}}{33}$$

Dus,

$$\int{x^{2} \left(x^{3} - 1\right)^{10} d x} = \frac{\left(x^{3} - 1\right)^{11}}{33}$$

Voeg de integratieconstante toe:

$$\int{x^{2} \left(x^{3} - 1\right)^{10} d x} = \frac{\left(x^{3} - 1\right)^{11}}{33}+C$$

$$$\int x^{2} \left(x^{3} - 1\right)^{10}\, dx = \frac{\left(x^{3} - 1\right)^{11}}{33} + C$$$A