Integral of $$$- 2 a + x^{4} - 19 x^{2} - 14 x + 32$$$ with respect to $$$x$$$

Find $$$\int \left(- 2 a + x^{4} - 19 x^{2} - 14 x + 32\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- 2 a + x^{4} - 19 x^{2} - 14 x + 32\right)d x}}} = {\color{red}{\left(\int{32 d x} - \int{2 a d x} - \int{14 x d x} - \int{19 x^{2} d x} + \int{x^{4} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=32$$$:

$$- \int{2 a d x} - \int{14 x d x} - \int{19 x^{2} d x} + \int{x^{4} d x} + {\color{red}{\int{32 d x}}} = - \int{2 a d x} - \int{14 x d x} - \int{19 x^{2} d x} + \int{x^{4} d x} + {\color{red}{\left(32 x\right)}}$$

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

$$32 x - \int{2 a d x} - \int{14 x d x} - \int{19 x^{2} d x} + {\color{red}{\int{x^{4} d x}}}=32 x - \int{2 a d x} - \int{14 x d x} - \int{19 x^{2} d x} + {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=32 x - \int{2 a d x} - \int{14 x d x} - \int{19 x^{2} d x} + {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=19$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$\frac{x^{5}}{5} + 32 x - \int{2 a d x} - \int{14 x d x} - {\color{red}{\int{19 x^{2} d x}}} = \frac{x^{5}}{5} + 32 x - \int{2 a d x} - \int{14 x d x} - {\color{red}{\left(19 \int{x^{2} d x}\right)}}$$

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

$$\frac{x^{5}}{5} + 32 x - \int{2 a d x} - \int{14 x d x} - 19 {\color{red}{\int{x^{2} d x}}}=\frac{x^{5}}{5} + 32 x - \int{2 a d x} - \int{14 x d x} - 19 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x^{5}}{5} + 32 x - \int{2 a d x} - \int{14 x d x} - 19 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=14$$$ and $$$f{\left(x \right)} = x$$$:

$$\frac{x^{5}}{5} - \frac{19 x^{3}}{3} + 32 x - \int{2 a d x} - {\color{red}{\int{14 x d x}}} = \frac{x^{5}}{5} - \frac{19 x^{3}}{3} + 32 x - \int{2 a d x} - {\color{red}{\left(14 \int{x d x}\right)}}$$

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

$$\frac{x^{5}}{5} - \frac{19 x^{3}}{3} + 32 x - \int{2 a d x} - 14 {\color{red}{\int{x d x}}}=\frac{x^{5}}{5} - \frac{19 x^{3}}{3} + 32 x - \int{2 a d x} - 14 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{5}}{5} - \frac{19 x^{3}}{3} + 32 x - \int{2 a d x} - 14 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=2 a$$$:

$$\frac{x^{5}}{5} - \frac{19 x^{3}}{3} - 7 x^{2} + 32 x - {\color{red}{\int{2 a d x}}} = \frac{x^{5}}{5} - \frac{19 x^{3}}{3} - 7 x^{2} + 32 x - {\color{red}{\left(2 a x\right)}}$$

Therefore,

$$\int{\left(- 2 a + x^{4} - 19 x^{2} - 14 x + 32\right)d x} = - 2 a x + \frac{x^{5}}{5} - \frac{19 x^{3}}{3} - 7 x^{2} + 32 x$$

Simplify:

$$\int{\left(- 2 a + x^{4} - 19 x^{2} - 14 x + 32\right)d x} = \frac{x \left(- 30 a + 3 x^{4} - 95 x^{2} - 105 x + 480\right)}{15}$$

Add the constant of integration:

$$\int{\left(- 2 a + x^{4} - 19 x^{2} - 14 x + 32\right)d x} = \frac{x \left(- 30 a + 3 x^{4} - 95 x^{2} - 105 x + 480\right)}{15}+C$$

$$$\int \left(- 2 a + x^{4} - 19 x^{2} - 14 x + 32\right)\, dx = \frac{x \left(- 30 a + 3 x^{4} - 95 x^{2} - 105 x + 480\right)}{15} + C$$$A