Integral de $$$\frac{\ln\left(x^{3}\right)}{\ln\left(2\right)}$$$

Encontre $$$\int \frac{3 \ln\left(x\right)}{\ln\left(2\right)}\, dx$$$.

A entrada é reescrita como: $$$\int{\frac{\ln{\left(x^{3} \right)}}{\ln{\left(2 \right)}} d x}=\int{\frac{3 \ln{\left(x \right)}}{\ln{\left(2 \right)}} d x}$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{3}{\ln{\left(2 \right)}}$$$ e $$$f{\left(x \right)} = \ln{\left(x \right)}$$$:

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

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

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

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

Assim,

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

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=1$$$:

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

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

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

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