Integral of $$$x^{7} \left(x^{8} - 3\right)^{33}$$$

Find $$$\int x^{7} \left(x^{8} - 3\right)^{33}\, dx$$$.

Let $$$u=x^{8} - 3$$$.

Then $$$du=\left(x^{8} - 3\right)^{\prime }dx = 8 x^{7} dx$$$ (steps can be seen »), and we have that $$$x^{7} dx = \frac{du}{8}$$$.

Therefore,

$${\color{red}{\int{x^{7} \left(x^{8} - 3\right)^{33} d x}}} = {\color{red}{\int{\frac{u^{33}}{8} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{8}$$$ and $$$f{\left(u \right)} = u^{33}$$$:

$${\color{red}{\int{\frac{u^{33}}{8} d u}}} = {\color{red}{\left(\frac{\int{u^{33} d u}}{8}\right)}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=33$$$:

$$\frac{{\color{red}{\int{u^{33} d u}}}}{8}=\frac{{\color{red}{\frac{u^{1 + 33}}{1 + 33}}}}{8}=\frac{{\color{red}{\left(\frac{u^{34}}{34}\right)}}}{8}$$

Recall that $$$u=x^{8} - 3$$$:

$$\frac{{\color{red}{u}}^{34}}{272} = \frac{{\color{red}{\left(x^{8} - 3\right)}}^{34}}{272}$$

Therefore,

$$\int{x^{7} \left(x^{8} - 3\right)^{33} d x} = \frac{\left(x^{8} - 3\right)^{34}}{272}$$

Add the constant of integration:

$$\int{x^{7} \left(x^{8} - 3\right)^{33} d x} = \frac{\left(x^{8} - 3\right)^{34}}{272}+C$$

$$$\int x^{7} \left(x^{8} - 3\right)^{33}\, dx = \frac{\left(x^{8} - 3\right)^{34}}{272} + C$$$A