Integral de $$$\ln\left(\sqrt{3} x\right)$$$

Encontre $$$\int \ln\left(\sqrt{3} x\right)\, dx$$$.

Seja $$$u=\sqrt{3} x$$$.

Então $$$du=\left(\sqrt{3} x\right)^{\prime }dx = \sqrt{3} dx$$$ (veja os passos »), e obtemos $$$dx = \frac{\sqrt{3} du}{3}$$$.

A integral pode ser reescrita como

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{\sqrt{3}}{3}$$$ e $$$f{\left(u \right)} = \ln{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\sqrt{3} \ln{\left(u \right)}}{3} d u}}} = {\color{red}{\left(\frac{\sqrt{3} \int{\ln{\left(u \right)} d u}}{3}\right)}}$$

Para a integral $$$\int{\ln{\left(u \right)} d u}$$$, use integração por partes $$$\int \operatorname{m} \operatorname{dv} = \operatorname{m}\operatorname{v} - \int \operatorname{v} \operatorname{dm}$$$.

Sejam $$$\operatorname{m}=\ln{\left(u \right)}$$$ e $$$\operatorname{dv}=du$$$.

Então $$$\operatorname{dm}=\left(\ln{\left(u \right)}\right)^{\prime }du=\frac{du}{u}$$$ (os passos podem ser vistos ») e $$$\operatorname{v}=\int{1 d u}=u$$$ (os passos podem ser vistos »).

A integral pode ser reescrita como

$$\frac{\sqrt{3} {\color{red}{\int{\ln{\left(u \right)} d u}}}}{3}=\frac{\sqrt{3} {\color{red}{\left(\ln{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{u} d u}\right)}}}{3}=\frac{\sqrt{3} {\color{red}{\left(u \ln{\left(u \right)} - \int{1 d u}\right)}}}{3}$$

Aplique a regra da constante $$$\int c\, du = c u$$$ usando $$$c=1$$$:

$$\frac{\sqrt{3} \left(u \ln{\left(u \right)} - {\color{red}{\int{1 d u}}}\right)}{3} = \frac{\sqrt{3} \left(u \ln{\left(u \right)} - {\color{red}{u}}\right)}{3}$$

Recorde que $$$u=\sqrt{3} x$$$:

$$\frac{\sqrt{3} \left(- {\color{red}{u}} + {\color{red}{u}} \ln{\left({\color{red}{u}} \right)}\right)}{3} = \frac{\sqrt{3} \left(- {\color{red}{\sqrt{3} x}} + {\color{red}{\sqrt{3} x}} \ln{\left({\color{red}{\sqrt{3} x}} \right)}\right)}{3}$$

Portanto,

$$\int{\ln{\left(\sqrt{3} x \right)} d x} = \frac{\sqrt{3} \left(\sqrt{3} x \ln{\left(\sqrt{3} x \right)} - \sqrt{3} x\right)}{3}$$

Simplifique:

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

Adicione a constante de integração:

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

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