Integral de $$$\frac{\ln\left(4^{y}\right)}{\ln\left(5\right)}$$$

Encontre $$$\int \frac{y \ln\left(4\right)}{\ln\left(5\right)}\, dy$$$.

A entrada é reescrita como: $$$\int{\frac{\ln{\left(4^{y} \right)}}{\ln{\left(5 \right)}} d y}=\int{\frac{y \ln{\left(4 \right)}}{\ln{\left(5 \right)}} d y}$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ usando $$$c=\frac{\ln{\left(4 \right)}}{\ln{\left(5 \right)}}$$$ e $$$f{\left(y \right)} = y$$$:

$${\color{red}{\int{\frac{y \ln{\left(4 \right)}}{\ln{\left(5 \right)}} d y}}} = {\color{red}{\frac{\ln{\left(4 \right)} \int{y d y}}{\ln{\left(5 \right)}}}}$$

Aplique a regra da potência $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=1$$$:

$$\frac{\ln{\left(4 \right)} {\color{red}{\int{y d y}}}}{\ln{\left(5 \right)}}=\frac{\ln{\left(4 \right)} {\color{red}{\frac{y^{1 + 1}}{1 + 1}}}}{\ln{\left(5 \right)}}=\frac{\ln{\left(4 \right)} {\color{red}{\left(\frac{y^{2}}{2}\right)}}}{\ln{\left(5 \right)}}$$

Portanto,

$$\int{\frac{y \ln{\left(4 \right)}}{\ln{\left(5 \right)}} d y} = \frac{y^{2} \ln{\left(4 \right)}}{2 \ln{\left(5 \right)}}$$

Simplifique:

$$\int{\frac{y \ln{\left(4 \right)}}{\ln{\left(5 \right)}} d y} = \frac{y^{2} \ln{\left(2 \right)}}{\ln{\left(5 \right)}}$$

Adicione a constante de integração:

$$\int{\frac{y \ln{\left(4 \right)}}{\ln{\left(5 \right)}} d y} = \frac{y^{2} \ln{\left(2 \right)}}{\ln{\left(5 \right)}}+C$$

$$$\int \frac{y \ln\left(4\right)}{\ln\left(5\right)}\, dy = \frac{y^{2} \ln\left(2\right)}{\ln\left(5\right)} + C$$$A