Integraal van $$$- x^{2} \ln\left(x\right)$$$

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

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

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

Voor de integraal $$$\int{x^{2} \ln{\left(x \right)} d x}$$$, gebruik partiële integratie $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Zij $$$\operatorname{u}=\ln{\left(x \right)}$$$ en $$$\operatorname{dv}=x^{2} dx$$$.

Dan $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (de stappen zijn te zien ») en $$$\operatorname{v}=\int{x^{2} d x}=\frac{x^{3}}{3}$$$ (de stappen zijn te zien »).

De integraal wordt

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

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{3}$$$ en $$$f{\left(x \right)} = x^{2}$$$:

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

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

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

Dus,

$$\int{\left(- x^{2} \ln{\left(x \right)}\right)d x} = - \frac{x^{3} \ln{\left(x \right)}}{3} + \frac{x^{3}}{9}$$

Vereenvoudig:

$$\int{\left(- x^{2} \ln{\left(x \right)}\right)d x} = \frac{x^{3} \left(1 - 3 \ln{\left(x \right)}\right)}{9}$$

Voeg de integratieconstante toe:

$$\int{\left(- x^{2} \ln{\left(x \right)}\right)d x} = \frac{x^{3} \left(1 - 3 \ln{\left(x \right)}\right)}{9}+C$$

$$$\int \left(- x^{2} \ln\left(x\right)\right)\, dx = \frac{x^{3} \left(1 - 3 \ln\left(x\right)\right)}{9} + C$$$A