Integral de $$$\frac{2 t}{\left(t - 3\right)^{2}}$$$

Encontre $$$\int \frac{2 t}{\left(t - 3\right)^{2}}\, dt$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ usando $$$c=2$$$ e $$$f{\left(t \right)} = \frac{t}{\left(t - 3\right)^{2}}$$$:

$${\color{red}{\int{\frac{2 t}{\left(t - 3\right)^{2}} d t}}} = {\color{red}{\left(2 \int{\frac{t}{\left(t - 3\right)^{2}} d t}\right)}}$$

Reescreva o numerador do integrando como $$$t=t - 3+3$$$ e decomponha a fração:

$$2 {\color{red}{\int{\frac{t}{\left(t - 3\right)^{2}} d t}}} = 2 {\color{red}{\int{\left(\frac{1}{t - 3} + \frac{3}{\left(t - 3\right)^{2}}\right)d t}}}$$

Integre termo a termo:

$$2 {\color{red}{\int{\left(\frac{1}{t - 3} + \frac{3}{\left(t - 3\right)^{2}}\right)d t}}} = 2 {\color{red}{\left(\int{\frac{3}{\left(t - 3\right)^{2}} d t} + \int{\frac{1}{t - 3} d t}\right)}}$$

Seja $$$u=t - 3$$$.

Então $$$du=\left(t - 3\right)^{\prime }dt = 1 dt$$$ (veja os passos »), e obtemos $$$dt = du$$$.

Portanto,

$$2 \int{\frac{3}{\left(t - 3\right)^{2}} d t} + 2 {\color{red}{\int{\frac{1}{t - 3} d t}}} = 2 \int{\frac{3}{\left(t - 3\right)^{2}} d t} + 2 {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$2 \int{\frac{3}{\left(t - 3\right)^{2}} d t} + 2 {\color{red}{\int{\frac{1}{u} d u}}} = 2 \int{\frac{3}{\left(t - 3\right)^{2}} d t} + 2 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Recorde que $$$u=t - 3$$$:

$$2 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} + 2 \int{\frac{3}{\left(t - 3\right)^{2}} d t} = 2 \ln{\left(\left|{{\color{red}{\left(t - 3\right)}}}\right| \right)} + 2 \int{\frac{3}{\left(t - 3\right)^{2}} d t}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ usando $$$c=3$$$ e $$$f{\left(t \right)} = \frac{1}{\left(t - 3\right)^{2}}$$$:

$$2 \ln{\left(\left|{t - 3}\right| \right)} + 2 {\color{red}{\int{\frac{3}{\left(t - 3\right)^{2}} d t}}} = 2 \ln{\left(\left|{t - 3}\right| \right)} + 2 {\color{red}{\left(3 \int{\frac{1}{\left(t - 3\right)^{2}} d t}\right)}}$$

Seja $$$u=t - 3$$$.

Então $$$du=\left(t - 3\right)^{\prime }dt = 1 dt$$$ (veja os passos »), e obtemos $$$dt = du$$$.

A integral pode ser reescrita como

$$2 \ln{\left(\left|{t - 3}\right| \right)} + 6 {\color{red}{\int{\frac{1}{\left(t - 3\right)^{2}} d t}}} = 2 \ln{\left(\left|{t - 3}\right| \right)} + 6 {\color{red}{\int{\frac{1}{u^{2}} d u}}}$$

Aplique a regra da potência $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=-2$$$:

$$2 \ln{\left(\left|{t - 3}\right| \right)} + 6 {\color{red}{\int{\frac{1}{u^{2}} d u}}}=2 \ln{\left(\left|{t - 3}\right| \right)} + 6 {\color{red}{\int{u^{-2} d u}}}=2 \ln{\left(\left|{t - 3}\right| \right)} + 6 {\color{red}{\frac{u^{-2 + 1}}{-2 + 1}}}=2 \ln{\left(\left|{t - 3}\right| \right)} + 6 {\color{red}{\left(- u^{-1}\right)}}=2 \ln{\left(\left|{t - 3}\right| \right)} + 6 {\color{red}{\left(- \frac{1}{u}\right)}}$$

Recorde que $$$u=t - 3$$$:

$$2 \ln{\left(\left|{t - 3}\right| \right)} - 6 {\color{red}{u}}^{-1} = 2 \ln{\left(\left|{t - 3}\right| \right)} - 6 {\color{red}{\left(t - 3\right)}}^{-1}$$

Portanto,

$$\int{\frac{2 t}{\left(t - 3\right)^{2}} d t} = 2 \ln{\left(\left|{t - 3}\right| \right)} - \frac{6}{t - 3}$$

Simplifique:

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

Adicione a constante de integração:

$$\int{\frac{2 t}{\left(t - 3\right)^{2}} d t} = \frac{2 \left(\left(t - 3\right) \ln{\left(\left|{t - 3}\right| \right)} - 3\right)}{t - 3}+C$$

$$$\int \frac{2 t}{\left(t - 3\right)^{2}}\, dt = \frac{2 \left(\left(t - 3\right) \ln\left(\left|{t - 3}\right|\right) - 3\right)}{t - 3} + C$$$A