Integral of $$$\left(y^{2} - 3\right)^{3}$$$

Find $$$\int \left(y^{2} - 3\right)^{3}\, dy$$$.

Expand the expression:

$${\color{red}{\int{\left(y^{2} - 3\right)^{3} d y}}} = {\color{red}{\int{\left(y^{6} - 9 y^{4} + 27 y^{2} - 27\right)d y}}}$$

Integrate term by term:

$${\color{red}{\int{\left(y^{6} - 9 y^{4} + 27 y^{2} - 27\right)d y}}} = {\color{red}{\left(- \int{27 d y} + \int{27 y^{2} d y} - \int{9 y^{4} d y} + \int{y^{6} d y}\right)}}$$

Apply the constant rule $$$\int c\, dy = c y$$$ with $$$c=27$$$:

$$\int{27 y^{2} d y} - \int{9 y^{4} d y} + \int{y^{6} d y} - {\color{red}{\int{27 d y}}} = \int{27 y^{2} d y} - \int{9 y^{4} d y} + \int{y^{6} d y} - {\color{red}{\left(27 y\right)}}$$

Apply the power rule $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=6$$$:

$$- 27 y + \int{27 y^{2} d y} - \int{9 y^{4} d y} + {\color{red}{\int{y^{6} d y}}}=- 27 y + \int{27 y^{2} d y} - \int{9 y^{4} d y} + {\color{red}{\frac{y^{1 + 6}}{1 + 6}}}=- 27 y + \int{27 y^{2} d y} - \int{9 y^{4} d y} + {\color{red}{\left(\frac{y^{7}}{7}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ with $$$c=9$$$ and $$$f{\left(y \right)} = y^{4}$$$:

$$\frac{y^{7}}{7} - 27 y + \int{27 y^{2} d y} - {\color{red}{\int{9 y^{4} d y}}} = \frac{y^{7}}{7} - 27 y + \int{27 y^{2} d y} - {\color{red}{\left(9 \int{y^{4} d y}\right)}}$$

Apply the power rule $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$\frac{y^{7}}{7} - 27 y + \int{27 y^{2} d y} - 9 {\color{red}{\int{y^{4} d y}}}=\frac{y^{7}}{7} - 27 y + \int{27 y^{2} d y} - 9 {\color{red}{\frac{y^{1 + 4}}{1 + 4}}}=\frac{y^{7}}{7} - 27 y + \int{27 y^{2} d y} - 9 {\color{red}{\left(\frac{y^{5}}{5}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ with $$$c=27$$$ and $$$f{\left(y \right)} = y^{2}$$$:

$$\frac{y^{7}}{7} - \frac{9 y^{5}}{5} - 27 y + {\color{red}{\int{27 y^{2} d y}}} = \frac{y^{7}}{7} - \frac{9 y^{5}}{5} - 27 y + {\color{red}{\left(27 \int{y^{2} d y}\right)}}$$

Apply the power rule $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$\frac{y^{7}}{7} - \frac{9 y^{5}}{5} - 27 y + 27 {\color{red}{\int{y^{2} d y}}}=\frac{y^{7}}{7} - \frac{9 y^{5}}{5} - 27 y + 27 {\color{red}{\frac{y^{1 + 2}}{1 + 2}}}=\frac{y^{7}}{7} - \frac{9 y^{5}}{5} - 27 y + 27 {\color{red}{\left(\frac{y^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(y^{2} - 3\right)^{3} d y} = \frac{y^{7}}{7} - \frac{9 y^{5}}{5} + 9 y^{3} - 27 y$$

Simplify:

$$\int{\left(y^{2} - 3\right)^{3} d y} = \frac{y \left(5 y^{6} - 63 y^{4} + 315 y^{2} - 945\right)}{35}$$

Add the constant of integration:

$$\int{\left(y^{2} - 3\right)^{3} d y} = \frac{y \left(5 y^{6} - 63 y^{4} + 315 y^{2} - 945\right)}{35}+C$$

$$$\int \left(y^{2} - 3\right)^{3}\, dy = \frac{y \left(5 y^{6} - 63 y^{4} + 315 y^{2} - 945\right)}{35} + C$$$A