Integral de $$$u \left(t - 1\right)$$$ em relação a $$$t$$$

Encontre $$$\int u \left(t - 1\right)\, dt$$$.

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

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

Integre termo a termo:

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

Aplique a regra da constante $$$\int c\, dt = c t$$$ usando $$$c=1$$$:

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

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

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

Portanto,

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

Simplifique:

$$\int{u \left(t - 1\right) d t} = \frac{t u \left(t - 2\right)}{2}$$

Adicione a constante de integração:

$$\int{u \left(t - 1\right) d t} = \frac{t u \left(t - 2\right)}{2}+C$$

$$$\int u \left(t - 1\right)\, dt = \frac{t u \left(t - 2\right)}{2} + C$$$A