Integral de $$$\frac{1}{6 - \frac{a}{50}}$$$

Encontre $$$\int \frac{1}{6 - \frac{a}{50}}\, da$$$.

Seja $$$u=6 - \frac{a}{50}$$$.

Então $$$du=\left(6 - \frac{a}{50}\right)^{\prime }da = - \frac{da}{50}$$$ (veja os passos »), e obtemos $$$da = - 50 du$$$.

Assim,

$${\color{red}{\int{\frac{1}{6 - \frac{a}{50}} d a}}} = {\color{red}{\int{\left(- \frac{50}{u}\right)d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=-50$$$ e $$$f{\left(u \right)} = \frac{1}{u}$$$:

$${\color{red}{\int{\left(- \frac{50}{u}\right)d u}}} = {\color{red}{\left(- 50 \int{\frac{1}{u} d u}\right)}}$$

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

$$- 50 {\color{red}{\int{\frac{1}{u} d u}}} = - 50 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Recorde que $$$u=6 - \frac{a}{50}$$$:

$$- 50 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = - 50 \ln{\left(\left|{{\color{red}{\left(6 - \frac{a}{50}\right)}}}\right| \right)}$$

Portanto,

$$\int{\frac{1}{6 - \frac{a}{50}} d a} = - 50 \ln{\left(\left|{\frac{a}{50} - 6}\right| \right)}$$

Simplifique:

$$\int{\frac{1}{6 - \frac{a}{50}} d a} = 50 \left(- \ln{\left(\left|{a - 300}\right| \right)} + \ln{\left(50 \right)}\right)$$

Adicione a constante de integração:

$$\int{\frac{1}{6 - \frac{a}{50}} d a} = 50 \left(- \ln{\left(\left|{a - 300}\right| \right)} + \ln{\left(50 \right)}\right)+C$$

$$$\int \frac{1}{6 - \frac{a}{50}}\, da = 50 \left(- \ln\left(\left|{a - 300}\right|\right) + \ln\left(50\right)\right) + C$$$A