Integral of $$$7 i^{4} - 12 i^{2}$$$

Find $$$\int \left(7 i^{4} - 12 i^{2}\right)\, di$$$.

Integrate term by term:

$${\color{red}{\int{\left(7 i^{4} - 12 i^{2}\right)d i}}} = {\color{red}{\left(- \int{12 i^{2} d i} + \int{7 i^{4} d i}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(i \right)}\, di = c \int f{\left(i \right)}\, di$$$ with $$$c=12$$$ and $$$f{\left(i \right)} = i^{2}$$$:

$$\int{7 i^{4} d i} - {\color{red}{\int{12 i^{2} d i}}} = \int{7 i^{4} d i} - {\color{red}{\left(12 \int{i^{2} d i}\right)}}$$

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

$$\int{7 i^{4} d i} - 12 {\color{red}{\int{i^{2} d i}}}=\int{7 i^{4} d i} - 12 {\color{red}{\frac{i^{1 + 2}}{1 + 2}}}=\int{7 i^{4} d i} - 12 {\color{red}{\left(\frac{i^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(i \right)}\, di = c \int f{\left(i \right)}\, di$$$ with $$$c=7$$$ and $$$f{\left(i \right)} = i^{4}$$$:

$$- 4 i^{3} + {\color{red}{\int{7 i^{4} d i}}} = - 4 i^{3} + {\color{red}{\left(7 \int{i^{4} d i}\right)}}$$

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

$$- 4 i^{3} + 7 {\color{red}{\int{i^{4} d i}}}=- 4 i^{3} + 7 {\color{red}{\frac{i^{1 + 4}}{1 + 4}}}=- 4 i^{3} + 7 {\color{red}{\left(\frac{i^{5}}{5}\right)}}$$

Therefore,

$$\int{\left(7 i^{4} - 12 i^{2}\right)d i} = \frac{7 i^{5}}{5} - 4 i^{3}$$

Simplify:

$$\int{\left(7 i^{4} - 12 i^{2}\right)d i} = \frac{i^{3} \left(7 i^{2} - 20\right)}{5}$$

Add the constant of integration:

$$\int{\left(7 i^{4} - 12 i^{2}\right)d i} = \frac{i^{3} \left(7 i^{2} - 20\right)}{5}+C$$

$$$\int \left(7 i^{4} - 12 i^{2}\right)\, di = \frac{i^{3} \left(7 i^{2} - 20\right)}{5} + C$$$A