Integraal van $$$\frac{\ln^{4}\left(x\right)}{2}$$$

Bepaal $$$\int \frac{\ln^{4}\left(x\right)}{2}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(x \right)} = \ln{\left(x \right)}^{4}$$$:

$${\color{red}{\int{\frac{\ln{\left(x \right)}^{4}}{2} d x}}} = {\color{red}{\left(\frac{\int{\ln{\left(x \right)}^{4} d x}}{2}\right)}}$$

Voor de integraal $$$\int{\ln{\left(x \right)}^{4} d x}$$$, gebruik partiële integratie $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Zij $$$\operatorname{u}=\ln{\left(x \right)}^{4}$$$ en $$$\operatorname{dv}=dx$$$.

Dan $$$\operatorname{du}=\left(\ln{\left(x \right)}^{4}\right)^{\prime }dx=\frac{4 \ln{\left(x \right)}^{3}}{x} dx$$$ (de stappen zijn te zien ») en $$$\operatorname{v}=\int{1 d x}=x$$$ (de stappen zijn te zien »).

Dus,

$$\frac{{\color{red}{\int{\ln{\left(x \right)}^{4} d x}}}}{2}=\frac{{\color{red}{\left(\ln{\left(x \right)}^{4} \cdot x-\int{x \cdot \frac{4 \ln{\left(x \right)}^{3}}{x} d x}\right)}}}{2}=\frac{{\color{red}{\left(x \ln{\left(x \right)}^{4} - \int{4 \ln{\left(x \right)}^{3} d x}\right)}}}{2}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=4$$$ en $$$f{\left(x \right)} = \ln{\left(x \right)}^{3}$$$:

$$\frac{x \ln{\left(x \right)}^{4}}{2} - \frac{{\color{red}{\int{4 \ln{\left(x \right)}^{3} d x}}}}{2} = \frac{x \ln{\left(x \right)}^{4}}{2} - \frac{{\color{red}{\left(4 \int{\ln{\left(x \right)}^{3} d x}\right)}}}{2}$$

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

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

Dan $$$\operatorname{du}=\left(\ln{\left(x \right)}^{3}\right)^{\prime }dx=\frac{3 \ln{\left(x \right)}^{2}}{x} dx$$$ (de stappen zijn te zien ») en $$$\operatorname{v}=\int{1 d x}=x$$$ (de stappen zijn te zien »).

Dus,

$$\frac{x \ln{\left(x \right)}^{4}}{2} - 2 {\color{red}{\int{\ln{\left(x \right)}^{3} d x}}}=\frac{x \ln{\left(x \right)}^{4}}{2} - 2 {\color{red}{\left(\ln{\left(x \right)}^{3} \cdot x-\int{x \cdot \frac{3 \ln{\left(x \right)}^{2}}{x} d x}\right)}}=\frac{x \ln{\left(x \right)}^{4}}{2} - 2 {\color{red}{\left(x \ln{\left(x \right)}^{3} - \int{3 \ln{\left(x \right)}^{2} d x}\right)}}$$

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

$$\frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 2 {\color{red}{\int{3 \ln{\left(x \right)}^{2} d x}}} = \frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 2 {\color{red}{\left(3 \int{\ln{\left(x \right)}^{2} d x}\right)}}$$

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

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

Dan $$$\operatorname{du}=\left(\ln{\left(x \right)}^{2}\right)^{\prime }dx=\frac{2 \ln{\left(x \right)}}{x} dx$$$ (de stappen zijn te zien ») en $$$\operatorname{v}=\int{1 d x}=x$$$ (de stappen zijn te zien »).

De integraal kan worden herschreven als

$$\frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 {\color{red}{\int{\ln{\left(x \right)}^{2} d x}}}=\frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 {\color{red}{\left(\ln{\left(x \right)}^{2} \cdot x-\int{x \cdot \frac{2 \ln{\left(x \right)}}{x} d x}\right)}}=\frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 {\color{red}{\left(x \ln{\left(x \right)}^{2} - \int{2 \ln{\left(x \right)} d x}\right)}}$$

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

$$\frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 x \ln{\left(x \right)}^{2} - 6 {\color{red}{\int{2 \ln{\left(x \right)} d x}}} = \frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 x \ln{\left(x \right)}^{2} - 6 {\color{red}{\left(2 \int{\ln{\left(x \right)} d x}\right)}}$$

Voor de integraal $$$\int{\ln{\left(x \right)} d x}$$$, 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 $$$\operatorname{dv}=dx$$$.

Dan $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (de stappen zijn te zien ») en $$$\operatorname{v}=\int{1 d x}=x$$$ (de stappen zijn te zien »).

Dus,

$$\frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 x \ln{\left(x \right)}^{2} - 12 {\color{red}{\int{\ln{\left(x \right)} d x}}}=\frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 x \ln{\left(x \right)}^{2} - 12 {\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}=\frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 x \ln{\left(x \right)}^{2} - 12 {\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=1$$$:

$$\frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 x \ln{\left(x \right)}^{2} - 12 x \ln{\left(x \right)} + 12 {\color{red}{\int{1 d x}}} = \frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 x \ln{\left(x \right)}^{2} - 12 x \ln{\left(x \right)} + 12 {\color{red}{x}}$$

Dus,

$$\int{\frac{\ln{\left(x \right)}^{4}}{2} d x} = \frac{x \ln{\left(x \right)}^{4}}{2} - 2 x \ln{\left(x \right)}^{3} + 6 x \ln{\left(x \right)}^{2} - 12 x \ln{\left(x \right)} + 12 x$$

Vereenvoudig:

$$\int{\frac{\ln{\left(x \right)}^{4}}{2} d x} = \frac{x \left(\ln{\left(x \right)}^{4} - 4 \ln{\left(x \right)}^{3} + 12 \ln{\left(x \right)}^{2} - 24 \ln{\left(x \right)} + 24\right)}{2}$$

Voeg de integratieconstante toe:

$$\int{\frac{\ln{\left(x \right)}^{4}}{2} d x} = \frac{x \left(\ln{\left(x \right)}^{4} - 4 \ln{\left(x \right)}^{3} + 12 \ln{\left(x \right)}^{2} - 24 \ln{\left(x \right)} + 24\right)}{2}+C$$

$$$\int \frac{\ln^{4}\left(x\right)}{2}\, dx = \frac{x \left(\ln^{4}\left(x\right) - 4 \ln^{3}\left(x\right) + 12 \ln^{2}\left(x\right) - 24 \ln\left(x\right) + 24\right)}{2} + C$$$A