Integraal van $$$\ln\left(x e^{8} - 9\right)$$$

Bepaal $$$\int \ln\left(x e^{8} - 9\right)\, dx$$$.

Zij $$$u=x e^{8} - 9$$$.

Dan $$$du=\left(x e^{8} - 9\right)^{\prime }dx = e^{8} dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{e^{8}}$$$.

Dus,

$${\color{red}{\int{\ln{\left(x e^{8} - 9 \right)} d x}}} = {\color{red}{\int{\frac{\ln{\left(u \right)}}{e^{8}} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=e^{-8}$$$ en $$$f{\left(u \right)} = \ln{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\ln{\left(u \right)}}{e^{8}} d u}}} = {\color{red}{\frac{\int{\ln{\left(u \right)} d u}}{e^{8}}}}$$

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

Zij $$$\operatorname{t}=\ln{\left(u \right)}$$$ en $$$\operatorname{dv}=du$$$.

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

Dus,

$$\frac{{\color{red}{\int{\ln{\left(u \right)} d u}}}}{e^{8}}=\frac{{\color{red}{\left(\ln{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{u} d u}\right)}}}{e^{8}}=\frac{{\color{red}{\left(u \ln{\left(u \right)} - \int{1 d u}\right)}}}{e^{8}}$$

Pas de constantenregel $$$\int c\, du = c u$$$ toe met $$$c=1$$$:

$$\frac{u \ln{\left(u \right)} - {\color{red}{\int{1 d u}}}}{e^{8}} = \frac{u \ln{\left(u \right)} - {\color{red}{u}}}{e^{8}}$$

We herinneren eraan dat $$$u=x e^{8} - 9$$$:

$$\frac{- {\color{red}{u}} + {\color{red}{u}} \ln{\left({\color{red}{u}} \right)}}{e^{8}} = \frac{- {\color{red}{\left(x e^{8} - 9\right)}} + {\color{red}{\left(x e^{8} - 9\right)}} \ln{\left({\color{red}{\left(x e^{8} - 9\right)}} \right)}}{e^{8}}$$

Dus,

$$\int{\ln{\left(x e^{8} - 9 \right)} d x} = \frac{- x e^{8} + \left(x e^{8} - 9\right) \ln{\left(x e^{8} - 9 \right)} + 9}{e^{8}}$$

Vereenvoudig:

$$\int{\ln{\left(x e^{8} - 9 \right)} d x} = \frac{\left(x e^{8} - 9\right) \left(\ln{\left(x e^{8} - 9 \right)} - 1\right)}{e^{8}}$$

Voeg de integratieconstante toe:

$$\int{\ln{\left(x e^{8} - 9 \right)} d x} = \frac{\left(x e^{8} - 9\right) \left(\ln{\left(x e^{8} - 9 \right)} - 1\right)}{e^{8}}+C$$

$$$\int \ln\left(x e^{8} - 9\right)\, dx = \frac{\left(x e^{8} - 9\right) \left(\ln\left(x e^{8} - 9\right) - 1\right)}{e^{8}} + C$$$A