Integral of $$$\frac{d}{t^{5}}$$$ with respect to $$$t$$$

Find $$$\int \frac{d}{t^{5}}\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=d$$$ and $$$f{\left(t \right)} = \frac{1}{t^{5}}$$$:

$${\color{red}{\int{\frac{d}{t^{5}} d t}}} = {\color{red}{d \int{\frac{1}{t^{5}} d t}}}$$

Apply the power rule $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-5$$$:

$$d {\color{red}{\int{\frac{1}{t^{5}} d t}}}=d {\color{red}{\int{t^{-5} d t}}}=d {\color{red}{\frac{t^{-5 + 1}}{-5 + 1}}}=d {\color{red}{\left(- \frac{t^{-4}}{4}\right)}}=d {\color{red}{\left(- \frac{1}{4 t^{4}}\right)}}$$

Therefore,

$$\int{\frac{d}{t^{5}} d t} = - \frac{d}{4 t^{4}}$$

Add the constant of integration:

$$\int{\frac{d}{t^{5}} d t} = - \frac{d}{4 t^{4}}+C$$

$$$\int \frac{d}{t^{5}}\, dt = - \frac{d}{4 t^{4}} + C$$$A