Integral de $$$u \left(t - 1\right)$$$ con respecto a $$$t$$$

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

Aplica la regla del factor constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ con $$$c=u$$$ y $$$f{\left(t \right)} = t - 1$$$:

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

Integra término a término:

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

Aplica la regla de la constante $$$\int c\, dt = c t$$$ con $$$c=1$$$:

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

Aplica la regla de la potencia $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=1$$$:

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

Por lo tanto,

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

Simplificar:

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

Añade la constante de integración:

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

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