Integral of $$$\left(x - 4\right)^{7}$$$

Find $$$\int \left(x - 4\right)^{7}\, dx$$$.

Let $$$u=x - 4$$$.

Then $$$du=\left(x - 4\right)^{\prime }dx = 1 dx$$$ (steps can be seen »), and we have that $$$dx = du$$$.

The integral becomes

$${\color{red}{\int{\left(x - 4\right)^{7} d x}}} = {\color{red}{\int{u^{7} d u}}}$$

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

$${\color{red}{\int{u^{7} d u}}}={\color{red}{\frac{u^{1 + 7}}{1 + 7}}}={\color{red}{\left(\frac{u^{8}}{8}\right)}}$$

Recall that $$$u=x - 4$$$:

$$\frac{{\color{red}{u}}^{8}}{8} = \frac{{\color{red}{\left(x - 4\right)}}^{8}}{8}$$

Therefore,

$$\int{\left(x - 4\right)^{7} d x} = \frac{\left(x - 4\right)^{8}}{8}$$

Add the constant of integration:

$$\int{\left(x - 4\right)^{7} d x} = \frac{\left(x - 4\right)^{8}}{8}+C$$

$$$\int \left(x - 4\right)^{7}\, dx = \frac{\left(x - 4\right)^{8}}{8} + C$$$A