Integral de $$$e a^{3} l^{3} t^{3} u v$$$ em relação a $$$t$$$

Encontre $$$\int e a^{3} l^{3} t^{3} u v\, dt$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ usando $$$c=e a^{3} l^{3} u v$$$ e $$$f{\left(t \right)} = t^{3}$$$:

$${\color{red}{\int{e a^{3} l^{3} t^{3} u v d t}}} = {\color{red}{e a^{3} l^{3} u v \int{t^{3} d t}}}$$

Aplique a regra da potência $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=3$$$:

$$e a^{3} l^{3} u v {\color{red}{\int{t^{3} d t}}}=e a^{3} l^{3} u v {\color{red}{\frac{t^{1 + 3}}{1 + 3}}}=e a^{3} l^{3} u v {\color{red}{\left(\frac{t^{4}}{4}\right)}}$$

Portanto,

$$\int{e a^{3} l^{3} t^{3} u v d t} = \frac{e a^{3} l^{3} t^{4} u v}{4}$$

Adicione a constante de integração:

$$\int{e a^{3} l^{3} t^{3} u v d t} = \frac{e a^{3} l^{3} t^{4} u v}{4}+C$$

$$$\int e a^{3} l^{3} t^{3} u v\, dt = \frac{e a^{3} l^{3} t^{4} u v}{4} + C$$$A