Integral de $$$1316141568000 \sqrt{7} \pi a^{4} b c^{2} t^{\frac{5}{2}} e^{2}$$$ em relação a $$$t$$$

Encontre $$$\int 1316141568000 \sqrt{7} \pi a^{4} b c^{2} t^{\frac{5}{2}} e^{2}\, dt$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ usando $$$c=1316141568000 \sqrt{7} \pi a^{4} b c^{2} e^{2}$$$ e $$$f{\left(t \right)} = t^{\frac{5}{2}}$$$:

$${\color{red}{\int{1316141568000 \sqrt{7} \pi a^{4} b c^{2} t^{\frac{5}{2}} e^{2} d t}}} = {\color{red}{\left(1316141568000 \sqrt{7} \pi a^{4} b c^{2} e^{2} \int{t^{\frac{5}{2}} d t}\right)}}$$

Aplique a regra da potência $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=\frac{5}{2}$$$:

$$1316141568000 \sqrt{7} \pi a^{4} b c^{2} e^{2} {\color{red}{\int{t^{\frac{5}{2}} d t}}}=1316141568000 \sqrt{7} \pi a^{4} b c^{2} e^{2} {\color{red}{\frac{t^{1 + \frac{5}{2}}}{1 + \frac{5}{2}}}}=1316141568000 \sqrt{7} \pi a^{4} b c^{2} e^{2} {\color{red}{\left(\frac{2 t^{\frac{7}{2}}}{7}\right)}}$$

Portanto,

$$\int{1316141568000 \sqrt{7} \pi a^{4} b c^{2} t^{\frac{5}{2}} e^{2} d t} = 376040448000 \sqrt{7} \pi a^{4} b c^{2} t^{\frac{7}{2}} e^{2}$$

Adicione a constante de integração:

$$\int{1316141568000 \sqrt{7} \pi a^{4} b c^{2} t^{\frac{5}{2}} e^{2} d t} = 376040448000 \sqrt{7} \pi a^{4} b c^{2} t^{\frac{7}{2}} e^{2}+C$$

$$$\int 1316141568000 \sqrt{7} \pi a^{4} b c^{2} t^{\frac{5}{2}} e^{2}\, dt = 376040448000 \sqrt{7} \pi a^{4} b c^{2} t^{\frac{7}{2}} e^{2} + C$$$A