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

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

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 $$$f{\left(x \right)} = x^{2} \ln{\left(3 x \right)}$$$:

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

Voor de integraal $$$\int{x^{2} \ln{\left(3 x \right)} d x}$$$, gebruik partiële integratie $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Zij $$$\operatorname{u}=\ln{\left(3 x \right)}$$$ en $$$\operatorname{dv}=x^{2} dx$$$.

Dan $$$\operatorname{du}=\left(\ln{\left(3 x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (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

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

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 $$$f{\left(x \right)} = x^{2}$$$:

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

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

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

Dus,

$$\int{\frac{x^{2} \ln{\left(3 x \right)}}{3} d x} = \frac{x^{3} \ln{\left(3 x \right)}}{9} - \frac{x^{3}}{27}$$

Vereenvoudig:

$$\int{\frac{x^{2} \ln{\left(3 x \right)}}{3} d x} = \frac{x^{3} \left(3 \ln{\left(x \right)} - 1 + 3 \ln{\left(3 \right)}\right)}{27}$$

Voeg de integratieconstante toe:

$$\int{\frac{x^{2} \ln{\left(3 x \right)}}{3} d x} = \frac{x^{3} \left(3 \ln{\left(x \right)} - 1 + 3 \ln{\left(3 \right)}\right)}{27}+C$$

$$$\int \frac{x^{2} \ln\left(3 x\right)}{3}\, dx = \frac{x^{3} \left(3 \ln\left(x\right) - 1 + 3 \ln\left(3\right)\right)}{27} + C$$$A