$$$1316141568000 \sqrt{7} \pi a^{4} b c^{2} t^{\frac{5}{2}} e^{2}$$$ 关于$$$t$$$的积分

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

对 $$$c=1316141568000 \sqrt{7} \pi a^{4} b c^{2} e^{2}$$$ 和 $$$f{\left(t \right)} = t^{\frac{5}{2}}$$$ 应用常数倍法则 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$：

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

应用幂法则 $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=\frac{5}{2}$$$：

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

因此，

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

加上积分常数：

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

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