Integral de $$$t e^{\frac{1}{2}}$$$

Halla $$$\int t e^{\frac{1}{2}}\, dt$$$.

Aplica la regla del factor constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ con $$$c=e^{\frac{1}{2}}$$$ y $$$f{\left(t \right)} = t$$$:

$${\color{red}{\int{t e^{\frac{1}{2}} d t}}} = {\color{red}{e^{\frac{1}{2}} \int{t d t}}}$$

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

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

Por lo tanto,

$$\int{t e^{\frac{1}{2}} d t} = \frac{t^{2} e^{\frac{1}{2}}}{2}$$

Añade la constante de integración:

$$\int{t e^{\frac{1}{2}} d t} = \frac{t^{2} e^{\frac{1}{2}}}{2}+C$$

$$$\int t e^{\frac{1}{2}}\, dt = \frac{t^{2} e^{\frac{1}{2}}}{2} + C$$$A