Integral de $$$\frac{1}{\frac{x}{2} - 450}$$$

Encontre $$$\int \frac{1}{\frac{x}{2} - 450}\, dx$$$.

Seja $$$u=\frac{x}{2} - 450$$$.

Então $$$du=\left(\frac{x}{2} - 450\right)^{\prime }dx = \frac{dx}{2}$$$ (veja os passos »), e obtemos $$$dx = 2 du$$$.

Assim,

$${\color{red}{\int{\frac{1}{\frac{x}{2} - 450} d x}}} = {\color{red}{\int{\frac{2}{u} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=2$$$ e $$$f{\left(u \right)} = \frac{1}{u}$$$:

$${\color{red}{\int{\frac{2}{u} d u}}} = {\color{red}{\left(2 \int{\frac{1}{u} d u}\right)}}$$

A integral de $$$\frac{1}{u}$$$ é $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$2 {\color{red}{\int{\frac{1}{u} d u}}} = 2 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Recorde que $$$u=\frac{x}{2} - 450$$$:

$$2 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 2 \ln{\left(\left|{{\color{red}{\left(\frac{x}{2} - 450\right)}}}\right| \right)}$$

Portanto,

$$\int{\frac{1}{\frac{x}{2} - 450} d x} = 2 \ln{\left(\left|{\frac{x}{2} - 450}\right| \right)}$$

Simplifique:

$$\int{\frac{1}{\frac{x}{2} - 450} d x} = 2 \ln{\left(\left|{x - 900}\right| \right)} - 2 \ln{\left(2 \right)}$$

Adicione a constante de integração (e remova a constante da expressão):

$$\int{\frac{1}{\frac{x}{2} - 450} d x} = 2 \ln{\left(\left|{x - 900}\right| \right)}+C$$

$$$\int \frac{1}{\frac{x}{2} - 450}\, dx = 2 \ln\left(\left|{x - 900}\right|\right) + C$$$A