Integral of $$$- t^{8} + e t^{7}$$$

Find $$$\int \left(- t^{8} + e t^{7}\right)\, dt$$$.

Integrate term by term:

$${\color{red}{\int{\left(- t^{8} + e t^{7}\right)d t}}} = {\color{red}{\left(- \int{t^{8} d t} + \int{e t^{7} d t}\right)}}$$

Apply the power rule $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=8$$$:

$$\int{e t^{7} d t} - {\color{red}{\int{t^{8} d t}}}=\int{e t^{7} d t} - {\color{red}{\frac{t^{1 + 8}}{1 + 8}}}=\int{e t^{7} d t} - {\color{red}{\left(\frac{t^{9}}{9}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=e$$$ and $$$f{\left(t \right)} = t^{7}$$$:

$$- \frac{t^{9}}{9} + {\color{red}{\int{e t^{7} d t}}} = - \frac{t^{9}}{9} + {\color{red}{e \int{t^{7} d t}}}$$

Apply the power rule $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=7$$$:

$$- \frac{t^{9}}{9} + e {\color{red}{\int{t^{7} d t}}}=- \frac{t^{9}}{9} + e {\color{red}{\frac{t^{1 + 7}}{1 + 7}}}=- \frac{t^{9}}{9} + e {\color{red}{\left(\frac{t^{8}}{8}\right)}}$$

Therefore,

$$\int{\left(- t^{8} + e t^{7}\right)d t} = - \frac{t^{9}}{9} + \frac{e t^{8}}{8}$$

Simplify:

$$\int{\left(- t^{8} + e t^{7}\right)d t} = t^{8} \left(- \frac{t}{9} + \frac{e}{8}\right)$$

Add the constant of integration:

$$\int{\left(- t^{8} + e t^{7}\right)d t} = t^{8} \left(- \frac{t}{9} + \frac{e}{8}\right)+C$$

$$$\int \left(- t^{8} + e t^{7}\right)\, dt = t^{8} \left(- \frac{t}{9} + \frac{e}{8}\right) + C$$$A