Integralen av $$$\ln\left(5 x^{2}\right)$$$

Bestäm $$$\int \ln\left(5 x^{2}\right)\, dx$$$.

För integralen $$$\int{\ln{\left(5 x^{2} \right)} d x}$$$, använd partiell integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Låt $$$\operatorname{u}=\ln{\left(5 x^{2} \right)}$$$ och $$$\operatorname{dv}=dx$$$.

Då gäller $$$\operatorname{du}=\left(\ln{\left(5 x^{2} \right)}\right)^{\prime }dx=\frac{2}{x} dx$$$ (stegen kan ses ») och $$$\operatorname{v}=\int{1 d x}=x$$$ (stegen kan ses »).

Integralen blir

$${\color{red}{\int{\ln{\left(5 x^{2} \right)} d x}}}={\color{red}{\left(\ln{\left(5 x^{2} \right)} \cdot x-\int{x \cdot \frac{2}{x} d x}\right)}}={\color{red}{\left(x \ln{\left(5 x^{2} \right)} - \int{2 d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=2$$$:

$$x \ln{\left(5 x^{2} \right)} - {\color{red}{\int{2 d x}}} = x \ln{\left(5 x^{2} \right)} - {\color{red}{\left(2 x\right)}}$$

Alltså,

$$\int{\ln{\left(5 x^{2} \right)} d x} = x \ln{\left(5 x^{2} \right)} - 2 x$$

Förenkla:

$$\int{\ln{\left(5 x^{2} \right)} d x} = x \left(2 \ln{\left(x \right)} - 2 + \ln{\left(5 \right)}\right)$$

Lägg till integrationskonstanten:

$$\int{\ln{\left(5 x^{2} \right)} d x} = x \left(2 \ln{\left(x \right)} - 2 + \ln{\left(5 \right)}\right)+C$$

$$$\int \ln\left(5 x^{2}\right)\, dx = x \left(2 \ln\left(x\right) - 2 + \ln\left(5\right)\right) + C$$$A