Integral of $$$- 8 x + \tan{\left(x \right)} \sec{\left(x \right)}$$$

Find $$$\int \left(- 8 x + \tan{\left(x \right)} \sec{\left(x \right)}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- 8 x + \tan{\left(x \right)} \sec{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{8 x d x} + \int{\tan{\left(x \right)} \sec{\left(x \right)} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=8$$$ and $$$f{\left(x \right)} = x$$$:

$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} - {\color{red}{\int{8 x d x}}} = \int{\tan{\left(x \right)} \sec{\left(x \right)} d x} - {\color{red}{\left(8 \int{x d x}\right)}}$$

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

$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} - 8 {\color{red}{\int{x d x}}}=\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} - 8 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} - 8 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

The integral of $$$\tan{\left(x \right)} \sec{\left(x \right)}$$$ is $$$\int{\tan{\left(x \right)} \sec{\left(x \right)} d x} = \sec{\left(x \right)}$$$:

$$- 4 x^{2} + {\color{red}{\int{\tan{\left(x \right)} \sec{\left(x \right)} d x}}} = - 4 x^{2} + {\color{red}{\sec{\left(x \right)}}}$$

Therefore,

$$\int{\left(- 8 x + \tan{\left(x \right)} \sec{\left(x \right)}\right)d x} = - 4 x^{2} + \sec{\left(x \right)}$$

Add the constant of integration:

$$\int{\left(- 8 x + \tan{\left(x \right)} \sec{\left(x \right)}\right)d x} = - 4 x^{2} + \sec{\left(x \right)}+C$$

$$$\int \left(- 8 x + \tan{\left(x \right)} \sec{\left(x \right)}\right)\, dx = \left(- 4 x^{2} + \sec{\left(x \right)}\right) + C$$$A