Integral of $$$a m x^{3} e^{- l}$$$ with respect to $$$a$$$

Find $$$\int a m x^{3} e^{- l}\, da$$$.

Apply the constant multiple rule $$$\int c f{\left(a \right)}\, da = c \int f{\left(a \right)}\, da$$$ with $$$c=m x^{3} e^{- l}$$$ and $$$f{\left(a \right)} = a$$$:

$${\color{red}{\int{a m x^{3} e^{- l} d a}}} = {\color{red}{m x^{3} e^{- l} \int{a d a}}}$$

Apply the power rule $$$\int a^{n}\, da = \frac{a^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=1$$$:

$$m x^{3} e^{- l} {\color{red}{\int{a d a}}}=m x^{3} e^{- l} {\color{red}{\frac{a^{1 + 1}}{1 + 1}}}=m x^{3} e^{- l} {\color{red}{\left(\frac{a^{2}}{2}\right)}}$$

Therefore,

$$\int{a m x^{3} e^{- l} d a} = \frac{a^{2} m x^{3} e^{- l}}{2}$$

Add the constant of integration:

$$\int{a m x^{3} e^{- l} d a} = \frac{a^{2} m x^{3} e^{- l}}{2}+C$$

$$$\int a m x^{3} e^{- l}\, da = \frac{a^{2} m x^{3} e^{- l}}{2} + C$$$A