Integraal van $$$\ln\left(a^{2} x^{2}\right)$$$ met betrekking tot $$$x$$$

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

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

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

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

Dus,

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

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=2$$$:

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

$$\int{\ln{\left(a^{2} x^{2} \right)} d x} = x \left(\ln{\left(a^{2} x^{2} \right)} - 2\right)+C$$

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