Integral de $$$t e^{5}$$$

Encontre $$$\int t e^{5}\, dt$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ usando $$$c=e^{5}$$$ e $$$f{\left(t \right)} = t$$$:

$${\color{red}{\int{t e^{5} d t}}} = {\color{red}{e^{5} \int{t d t}}}$$

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

$$e^{5} {\color{red}{\int{t d t}}}=e^{5} {\color{red}{\frac{t^{1 + 1}}{1 + 1}}}=e^{5} {\color{red}{\left(\frac{t^{2}}{2}\right)}}$$

Portanto,

$$\int{t e^{5} d t} = \frac{t^{2} e^{5}}{2}$$

Adicione a constante de integração:

$$\int{t e^{5} d t} = \frac{t^{2} e^{5}}{2}+C$$

$$$\int t e^{5}\, dt = \frac{t^{2} e^{5}}{2} + C$$$A