Integral de $$$t \sin^{2}{\left(\omega \right)}$$$ con respecto a $$$t$$$

Halla $$$\int t \sin^{2}{\left(\omega \right)}\, dt$$$.

Aplica la regla del factor constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ con $$$c=\sin^{2}{\left(\omega \right)}$$$ y $$$f{\left(t \right)} = t$$$:

$${\color{red}{\int{t \sin^{2}{\left(\omega \right)} d t}}} = {\color{red}{\sin^{2}{\left(\omega \right)} \int{t d t}}}$$

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

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

Por lo tanto,

$$\int{t \sin^{2}{\left(\omega \right)} d t} = \frac{t^{2} \sin^{2}{\left(\omega \right)}}{2}$$

Añade la constante de integración:

$$\int{t \sin^{2}{\left(\omega \right)} d t} = \frac{t^{2} \sin^{2}{\left(\omega \right)}}{2}+C$$

$$$\int t \sin^{2}{\left(\omega \right)}\, dt = \frac{t^{2} \sin^{2}{\left(\omega \right)}}{2} + C$$$A