Integral of $$$u \left(t - 1\right)$$$ with respect to $$$t$$$

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

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=u$$$ and $$$f{\left(t \right)} = t - 1$$$:

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

Integrate term by term:

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

Apply the constant rule $$$\int c\, dt = c t$$$ with $$$c=1$$$:

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

Apply the power rule $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=1$$$:

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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